On the topology of manifolds with positive intermediate curvature

Liam Mazurowski Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, 18015, United States [email protected] , Tongrui Wang School of Mathematical Sciences, Shanghai Jiao Tong University, Minhang District, Shanghai 200240, China [email protected] and Xuan Yao Department of Mathematics, Cornell University, Ithaca, New York, 14853, United States [email protected]

Abstract.

We formulate a conjecture relating the topology of a manifold’s universal cover with the existence of metrics with positive $m$ -intermediate curvature. We prove the result for manifolds of dimension $n\in\{3,4,5\}$ and for most choices of $m$ when $n=6$ . As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive $4$ -intermediate curvature.

1. Introduction

A manifold $M$ is called aspherical if the universal cover of $M$ is contractible. Equivalently, $M$ is aspherical if $\pi_{k}(M)=0$ for $k\geq 2$ . Thus an aspherical manifold $M$ is a $K(\pi,1)$ space for $\pi=\pi_{1}(M)$ . Schoen and Yau [16] conjectured that a closed aspherical manifold cannot carry a metric of positive scalar curvature; also see Gromov [6]. This $K(\pi,1)$ conjecture is known to be true when $n=3$ by work of Schoen-Yau [15] and Gromov-Lawson [7], when $n=4$ by Chodosh-Li [3], and when $n=5$ by Chodosh-Li [3] and independently by Gromov [10].

Theorem 1.

Assume that $M^{n}$ is a closed $K(\pi,1)$ manifold of dimension $n\in\{3,4,5\}$ . Then $M$ does not admit a metric with positive scalar curvature.

Let $T^{n}$ be the $n$ -dimensional torus. The Geroch conjecture states that $T^{n}$ does not admit a metric of positive scalar curvature. This can be seen as a special case of the $K(\pi,1)$ conjecture. The Geroch conjecture was resolved by Schoen-Yau [15] for $n\leq 7$ and by Gromov-Lawson [8] in all dimensions. In a different direction, Brendle-Hirsch-Johne [2] have generalized the Geroch conjecture to show that a closed manifold $M^{n}=N^{n-m}\times T^{m}$ does not admit a metric with positive $m$ -intermediate curvature when $n\leq 7$ . Here the $m$ -intermediate curvature (see Definition 8) interpolates between the Ricci curvature ( $m=1$ ) and the scalar curvature ( $m=n-1$ ).

Theorem 2 (Brendle-Hirsch-Johne [2]).

A closed manifold $M^{n}=N^{n-m}\times T^{m}$ does not admit a metric with positive $m$ -intermediate curvature when $n\leq 7$ .

In this paper, we propose the following conjecture which relates the topology of a manifold’s universal cover with the existence of metrics with positive $m$ -intermediate curvature. We note that both Theorem 1 and Theorem 2 occur as special cases of this conjecture.

Conjecture 3.

Let $M^{n}$ be a closed manifold of dimension $3\leq n\leq 7$ . Assume that the universal cover $\overline{M}$ of $M$ satisfies

H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=\ldots=H_{n-m+1}(\overline{M},\mathbb{Z})=0.

Then $M$ does not admit a metric with positive $m$ -intermediate curvature.

As our main result, we show that Conjecture 3 is true for $n\in\{3,4,5\}$ and for most choices of $m$ when $n=6$ .

Theorem 4 (Main Theorem).

Suppose $M^{n}$ is a closed manifold and that the universal cover $\overline{M}$ of $M$ satisfies

H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=\ldots=H_{n-m+1}(\overline{M},\mathbb{Z})=0.

Then $M$ does not admit a metric of positive $m$ -intermediate curvature in any of the following cases: $n\in\{3,4,5\}$ and $m\in\{1,2,\dots,n-1\}$ ; $n=6$ and $m\in\{1,2,3,4\}$ ; $n\geq 7$ and $m=1$ .

In dimension $n=6$ , the $5$ -intermediate curvature is equivalent to the scalar curvature, and positive $4$ -intermediate curvature is a slight strengthening of positive scalar curvature. As a corollary of the $n=6$ and $m=4$ case of our main theorem, we see that a closed, aspherical $6$ -manifold cannot admit a metric with positive $4$ -intermediate curvature. This provides some evidence for the $K(\pi,1)$ conjecture in dimension 6.

Corollary 5.

A closed, aspherical $6$ -manifold does not admit a metric with positive 4-intermediate curvature.

Finally, we note that Chodosh-Li-Liokumovich [4] have proven a mapping version of the $K(\pi,1)$ conjecture and a reformulation of the $K(\pi,1)$ conjecture as a classification theorem. We show that corresponding results also hold in our setting.

Theorem 6.

Suppose $M^{n}$ is a closed manifold and that the universal cover $\overline{M}$ of $M$ satisfies

H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=\ldots=H_{n-m+1}(\overline{M},\mathbb{Z})=0.

Assume further that $N^{n}$ is a closed manifold with a non-zero degree map to $M^{n}$ . Then $N$ does not admit a metric of positive $m$ -intermediate curvature in any of the following cases: $n\in\{3,4,5\}$ and $m\in\{1,2,\dots,n-1\}$ ; $n=6$ and $m\in\{1,2,3,4\}$ ; $n\geq 7$ and $m=1$ .

Theorem 7.

Suppose $M^{n}$ is a closed manifold which admits positive $m$ -intermediate curvature. Suppose further that

(1)

$n=5$ , $m=2$ , and $\pi_{2}(M^{5})=0$ ; or
(2)

$n=6$ , $m=4$ , and $\pi_{2}(M^{6})=\pi_{3}(M^{6})=\pi_{4}(M^{6})=0$ .

Then a finite covering of $N$ is homeomorphic to $S^{n}$ or connected sums of $S^{n-1}\times S^{1}$ .

1.1. Further Discussion and Proof Ideas

Schoen and Yau’s proof [15] of the Geroch conjecture uses minimal hypersurfaces and an inductive descent argument. A key step in the argument is to show that if $(M,g)$ has positive scalar curvature and $\Sigma$ is an area minimizing hypersurface in $M$ then $\Sigma$ admits a conformal metric with positive scalar curvature. In fact, the necessary conformal factor is a suitable power of the first eigenfunction for the second variation of area.

Brendle, Hirsch, and Johne [2] proved a generalization of the Geroch conjecture to closed manifolds of the form $M^{n}=N^{n-m}\times T^{m}$ . To do so, they introduced a new curvature condition called $m$ -intermediate curvature.

Definition 8.

Suppose $(M^{n},g)$ is Riemannian manifold. Given a collection of orthonormal vectors $\{e_{1},\cdots,e_{m}\}$ at a point $p\in M$ , let $\{e_{1},\cdots e_{m},e_{m+1},\ldots,e_{n}\}$ be an extension to an orthonormal basis of $T_{p}M$ . The $m$ -intermediate curvature $C_{m}$ of the orthonormal vectors $\{e_{1},\cdots,e_{m}\}$ is defined by

C_{m}(e_{1},\cdots,e_{m}):=\sum_{p=1}^{m}\sum_{q=m+1}^{n}R_{M}(e_{p},e_{q},e_{p},e_{q}).

We say that $(M,g)$ has positive $m$ -intermediate curvature if for any choice of orthonormal vectors $\{e_{1},\cdots,e_{m}\}$ at any point $p\in M$ , we have $C_{m}(e_{1},\cdots,e_{m})>0$ .

We note that the 1-intermediate curvature is precisely the Ricci curvature and that the $(n-1)$ -intermediate curvature is equal to the scalar curvature up to a constant factor. The 2-intermediate curvature had previously been introduced and studied by Shen and Ye [17] as bi-Ricci curvature.

Brendle, Hirsch, and Johne’s main result says that $M^{n}=N^{n-m}\times T^{m}$ cannot admit a metric with positive $m$ -intermediate curvature when $n\leq 7$ . Their proof is based on the fact that certain weighted minimal slicings do not exist in manifolds with positive $m$ -intermediate curvature. In the construction of their weighted slicings, each successive weight $\rho_{k+1}$ is obtained from the previous weight $\rho_{k}$ by multiplying by the first eigenfunction for the second variation of $\rho_{k}$ -weighted area. This is reminiscent of the Schoen-Yau descent argument. In dimension $n>7$ , Xu [19] constructed interesting counter-examples to Brendle-Hirsch-Johne’s generalization of the Geroch Conjecture, which is the reason why we only expect Conjecture 3 to be true for $3\leq n\leq 7$ .

Recall that the Geroch conjecture is a particular case of the $K(\pi,1)$ conjecture. In their 1987 paper, Schoen and Yau [16] put forth an outline for how to prove the $K(\pi,1)$ conjecture in dimension $n=4$ . Chodosh and Li’s proof [3] is inspired by this outline. Let $M^{n}$ be a closed, aspherical manifold. Assume for contradiction that $M$ has a metric with positive scalar curvature. Working in the universal cover $\overline{M}$ of $M$ , Chodosh and Li construct a geodesic line $\sigma$ and a closed null-homologous submanifold $\Lambda^{n-2}$ which is far away from $\sigma$ but linked with $\sigma$ . Let $\Sigma_{1}$ be the area minimizing filling of $\Lambda$ . Using a clever choice of prescribed mean curvature functional, Chodosh and Li construct a $\mu$ -bubble $\Sigma_{2}\subset\Sigma_{1}$ such that $\Lambda$ is homologous to $\Sigma_{2}$ in $\Sigma_{1}$ and $\Sigma_{2}$ lies far away from $\sigma$ . When $n=4$ , the $\mu$ -bubble $\Sigma_{2}$ is 2-dimensional and it is possible to show with a second variation argument that each component of $\Sigma_{2}$ has uniformly bounded diameter. Thus, by a general property of universal covers, it is possible to fill each component of $\Sigma_{2}$ within a fixed sized neighborhood in $\overline{M}$ . In this manner, one obtains a new filling of $\Lambda$ which does not intersect $\sigma$ and this is a contradiction.

When $n=5$ , new difficulties arise because the $\mu$ -bubble $\Sigma_{2}$ is 3-dimensional and may not have a diameter bound. To overcome this, Chodosh and Li developed the so-called slice-and-dice argument. They first slice $\Sigma_{2}$ open along suitable $\mu$ -bubbles $\Sigma_{3,1},\ldots,\Sigma_{3,k}$ to obtain a manifold $\hat{\Sigma}_{2}$ with simple 2 dimensional homology. Then they dice $\hat{\Sigma}_{2}$ into pieces with small diameter using free boundary $\mu$ -bubbles. These free boundary $\mu$ -bubbles are two dimensional, so it is possible to use a second variation argument together with the Gauss-Bonnet theorem to show that each dicing surface is a disk. Crucially, the boundary of a disk is connected, and this gives a simple combinatorial structure to the output of the slice and dice procedure. This can then be used to construct a filling of $\Sigma_{2}$ within a fixed sized neighborhood in $\overline{M}$ as before.

Our proof of Theorem 4 is based on the argument of Chodosh and Li. The main observation is that positive $m$ -intermediate curvature should imply a diameter bound on the slice $\Sigma_{m-1}$ in low dimensions. A key tool in this regard is the generalized Bonnet-Myers theorem discovered by Shen and Ye [18]. We will apply the generalized Bonnet-Myers theorem with a weight equal to a product of eigenfunctions for the stability operators of certain weighted area and $\mu$ -bubble functionals. We remark that the case $n=6$ and $m=3$ is very delicate and requires the full strength of the Shen-Ye generalized Bonnet-Myers theorem as well as the use of a carefully chosen weighted $\mu$ -bubble functional.

To handle the case $n=6$ and $m=4$ , we adapt the slice and dice procedure of Chodosh and Li. Since the dicing submanifolds are now 3-dimensional, we can no longer use the Gauss-Bonnet theorem to determine their topology. Nevertheless, we can prove the following Frankel type theorem which implies that the dicing submanifolds have connected boundary:

Theorem 9.

Let $\Sigma$ be a $3$ -dimensional compact Riemannian manifold with boundary. Assume $\Sigma$ admits a positive function $f$ which satisfies

\operatorname{Ric}(v,v)-f^{-1}\Delta f+\frac{1}{2}|\nabla\ln f|^{2}\geq 0

for all unit vectors $v$ . Moreover, suppose that $\partial_{\eta}f=-fH_{\partial\Sigma}$ where $\eta$ is the unit outward normal to $\partial\Sigma$ . Then $\partial\Sigma$ is connected.

The fact that the dicing submanifolds have connected boundary is then enough to carry through the remainder of the argument.

Remark 10.

The hypothesis of Shen and Ye’s theorem is dimension dependent, and we were not able to verify the hypothesis on the slice $\Sigma_{m-1}$ when $n=7$ and $m\in\{2,3,4\}$ .

1.2. Organization

In Section 2, we first introduce some convenient terminology and then we recall basic facts about $\mu$ -bubbles, including general existence and stability results. We prove a Frankel type theorem for positive conformal Ricci curvature in Section 3. Combining the weighted slicing techniques, Shen-Ye’s diameter estimate, and the slice-and-dice procedure of Chodosh-Li, we prove the Main Theorem in Section 4. In Section 5, we generalize our results to a mapping version and prove a refinement of the Main Theorem into a positive result.

1.3. Acknowledgments

We would like to thank Xin Zhou for his helpful discussions and guidance on this project. L.M. acknowledges the support of an AMS-Simons travel grant.

2. Preliminaries

In this section, we discuss some preliminary results that will be needed in the proof of our main theorem. First, we introduce some convenient terminology for manifolds where the homology of the universal cover vanishes in certain degrees.

Definition 11.

We say a closed manifold $M^{n}$ is $m$ -acyclic, $m\in\{1,2\cdots,n-1\}$ , if the universal cover $\overline{M}$ of $M$ satisfies

H_{n-m+1}(\overline{M};\mathbb{Z})=\cdots=H_{n}(\overline{M};\mathbb{Z})=0.

2.1. Topological Preliminaries

In this subsection, we discuss some topological results about the universal cover of a closed manifold.

Lemma 12.

Suppose $(M^{n},g)$ is an $m$ -acyclic closed manifold for some $m\in\{1,2,\cdots,n-1\}$ . Then there exists a geodesic line on $(\overline{M},\bar{g})$ , where $\bar{g}$ is the lifted Riemannian metric.

Proof.

Since $H_{n}(\overline{M};\mathbb{Z})=0$ , $\overline{M}$ is non-compact. The proof follows easily as in [3, Lemma 6]. ∎

More generally, we recall the following construction of Chodosh and Li [3].

Proposition 13.

Assume that $\overline{M}$ is the universal cover of a closed Riemannian manifold $M^{n}$ . Further suppose that $H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=0$ . Then for any $L>0$ , there exists a geodesic line $\sigma$ in $\overline{M}$ together with a closed, null-homologous manifold $\Lambda^{n-2}$ embedded in $\overline{M}$ such that

(i)

$\Lambda$ is linked with $\sigma$ , i.e., if $\Lambda=\partial\Sigma$ then $\Sigma$ has non-zero algebraic intersection number with $\sigma$ ,
(ii)

$d(\Lambda,\sigma)\geq L$ .

Proof.

This follows from the argument in [3, Section 2]. While Chodosh and Li consider the universal cover of a closed aspherical manifold, it is easy to see that their arguments apply to the universal cover $\overline{M}$ of an arbitrary closed manifold $M^{n}$ provided that $H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=0$ . ∎

Proposition 14.

Suppose $M^{n}$ is $m$ -acyclic. For any $r>0$ , there exists $R(r)$ with the following property. For any $k\geq n-m+1$ and any $k$ -cycle $\alpha$ with $\alpha\subset B_{r}(p)$ , we have $\alpha=\partial\beta$ with $\beta\subset B_{R}(p)$ .

Proof.

The proof is exactly the same as in Chodosh-Li [3]; see also He-Zhu [12]. ∎

2.2. Preliminaries on $\mu$ -bubbles

Next we recall some basic facts about $\mu$ -bubbles, including general existence and stability results.

For $n\leq 7$ , consider a Riemannian manifold $(M^{n},g)$ with boundary $\partial M=\partial_{-}M\sqcup\partial_{+}M$ , where neither of $\partial_{\pm}M$ is empty. Suppose $u$ is a smooth positive function on $M$ , and $h$ is a smooth function defined on the interior of $M$ so that $h\to\pm\infty$ as $x\to\partial_{\pm}M$ respectively. Given a Caccioppoli set $\Omega_{0}\subset M$ with smooth boundary and containing $\partial_{+}M$ , consider the $\mu$ -bubble functional:

(1)

\displaystyle\mathcal{A}(\Omega):=\int_{\partial^{*}\Omega}uda-\int_{M}(\chi_{\Omega}-\chi_{\Omega_{0}})hudv,

for all Caccioppoli set $\Omega\subset M$ with $\Omega\Delta\Omega_{0}\subset\subset\operatorname{int}(M)$ , where $\partial^{*}\Omega\subset\operatorname{int}(M)$ is the reduced boundary of $\Omega$ in $\operatorname{int}(M)$ . We call the $\mathcal{A}$ -minimizer $\Omega$ in this class a $\mu$ -bubble.

The existence and regularity of a minimizer of $\mathcal{A}$ among all Caccioppoli sets was claimed by Gromov [9] and proven rigorously by Zhu [20].

Proposition 15 (Gromov[9], Zhu [20]).

There exists a smooth minimizer $\Omega$ for $\mathcal{A}$ such that $\Omega\Delta\Omega_{0}\subset\subset\operatorname{int}(M^{n})$ , for $3\leq n\leq 7$ .

In Chodosh-Li’s proof [3] of the $K(\pi,1)$ conjecture in dimension $4$ and $5$ , they generalized the $\mu$ -bubble techniques to a free boundary version. In their setting, they assume that $(M^{n},g)$ is a Riemannian manifold with co-dimension $2$ corners in the sense that any boundary point has a neighborhood diffeomorphic to one of the following: $\{x\in\mathbb{R}^{n}:x_{n}\geq 0\}$ or $\{x\in\mathbb{R}^{n}:x_{n-1},x_{n}\geq 0\}$ . Furthermore, $\partial M=\partial_{\pm}M\cup\partial_{0}M$ , $\partial_{\pm}M$ meets $\partial_{0}M$ orthogonally, and $\partial_{\pm}M\cap\partial_{0}M$ consists of smooth co-dimension $2$ closed submanifolds. In the free boundary setting, we assume that

H_{\partial_{0}M}+u^{-1}\langle\nabla_{M}u,\nu_{\partial_{0}M}\rangle=0.

For the functional $\mathcal{A}$ defined as before, Chodosh-Li used similar arguments as in Zhu’s work [20] and proved the following results.

Proposition 16 (Chodosh-Li [3]).

Suppose $n\leq 7$ , there exists $\Omega$ with $\partial\Omega\subset\operatorname{int}(M)\cup\partial_{0}M$ minimizing $\mathcal{A}$ among such regions. The boundary $\partial\Omega$ is smooth and meets $\partial_{0}M$ orthogonally. We have

H=-u^{-1}\langle\nabla_{M}u,\nu_{\partial\Omega}\rangle+h

along $\partial\Omega$ , where $H$ is the mean curvature of $\partial\Omega$ with respect to the unit outer normal $\nu_{\partial\Omega}$ . Finally, if $\Sigma$ is a component of $\partial\Omega$ , then for any $\psi\in C^{1}(\Sigma)$ , we have

	$\displaystyle 0$	$\displaystyle\leq\left.\frac{d^{2}}{dt^{2}}\right\|_{t=0}\mathcal{A}(\Omega^{t})$
		$\displaystyle=\int_{\Sigma}-u\psi\Delta_{\Sigma}\psi-\left(\\|A_{\Sigma}\\|^{2}+\operatorname{Ric}_{M}(\nu,\nu)\right)u\psi^{2}-\psi\langle\nabla_{\Sigma}u,\nabla_{\Sigma}\psi\rangle$
		$\displaystyle\qquad\qquad+\psi^{2}\nabla^{2}_{M}u(\nu,\nu)-\psi^{2}u^{-1}\langle\nabla_{M}u,\nu\rangle^{2}-u\psi^{2}\langle\nabla_{M}h,\nu\rangle~{}d\mathcal{H}^{n-1}$
		$\displaystyle\qquad\qquad+\int_{\partial\Sigma}u\psi\frac{\partial\psi}{\partial\eta}-A_{\partial_{0}M}(\nu_{\partial\Sigma},\nu_{\partial\Sigma})u\psi^{2}~{}d\mathcal{H}^{n-2},$

where $\nu=\nu_{\partial\Omega}$ is the unit outer normal of $\Omega$ , $\eta$ is the unit co-normal of $\Sigma$ along $\partial\Sigma$ , and $\{\Omega^{t}\}$ is a smooth family of regions with $\Omega^{0}=\Omega$ and normal speed $\psi$ at $t=0$ .

2.3. Diameter Estimates

The classical Bonnet-Myers theorem gives a diameter bound for manifolds with positive Ricci curvature. Shen and Ye discovered the following diameter bound for stable minimal hypersurfaces in manifolds with positive 2-intermediate curvature.

Theorem 17 (Shen-Ye [17]).

Let $M^{n}$ be a complete Riemannian manifold of dimension $n\in\{3,4,5\}$ . Assume that the $2$ -intermediate curvature of $M$ is at least $\kappa>0$ . Let $\Sigma$ be a stable minimal hypersurface in $M$ . Then for every $p\in\Sigma$ , one has

d(p,\partial\Sigma)\leq\sqrt{c(n)}\frac{\pi}{\sqrt{\kappa}},

where $c(n)$ is a dimensional constant.

Shen and Ye also proved the following generalized Bonnet-Myers theorem which will be very useful for proving diameter bounds for certain weighted minimal slicings. We state the 3-dimensional version first since it is simpler and more powerful.

Theorem 18 (Shen-Ye [18]).

Let $N^{3}$ be a complete Riemannian manifold. Assume there is a function $f>0$ on $N$ such that

(2)

\operatorname{Ric}_{N}(e,e)-f^{-1}\Delta_{N}f+\frac{1}{2}|\nabla_{N}\ln f|^{2}\geq\kappa>0

for all unit tangent vectors $e$ . Then

\operatorname{diam}(N)\leq\sqrt{2}\frac{\pi}{\sqrt{\kappa}}.

In particular, $N$ is compact.

Definition 19.

Following Shen-Ye, we will say a manifold satisfying (2) admits positive conformal Ricci curvature.

In higher dimensions, the theorem must be modified as follows.

Theorem 20 (Shen-Ye [18]).

Let $N^{k}$ be a complete Riemannian manifold of dimension $k\geq 4$ . Assume there is a function $f>0$ on $N$ and $\tau,\varepsilon>0$ so that

\operatorname{Ric}_{N}(e,e)-\tau f^{-1}\Delta_{N}f+\left[\tau-\left(\frac{k-1}{4}+\varepsilon\right)\tau^{2}\right]|\nabla_{N}\ln f|^{2}\geq\kappa>0

for all unit tangent vectors $e$ . Then the diameter of $N$ is bounded above in terms of $k$ , $\kappa$ , and $\varepsilon$ .

3. Frankel Type Theorems

Recall that Frankel’s theorem says that if $M$ has positive Ricci curvature and minimal boundary then $\partial M$ is connected. In this section, we are going to prove a Frankel type theorem for manifolds with positive conformal Ricci curvature. As motivation, we first prove the following version of Frankel’s theorem for bi-Ricci curvature. This theorem fits into the general theme, first observed by Shen-Ye [17], that stable minimal hypersurfaces in manifolds with positive bi-Ricci curvature behave as if they had positive Ricci curvature. Curiously, in contrast to the diameter bounds, the following theorem does not require any dimension restriction.

Theorem 21.

Let $M^{n+1}$ be a compact manifold with boundary. Assume that $M$ has positive bi-Ricci curvature and minimal boundary. Assume that $\Sigma^{n}$ is a two-sided, stable, free boundary minimal hypersurface in $M$ . Then $\partial\Sigma$ is connected.

Proof.

Let $\nu$ be the unit normal vector to $\Sigma$ in $M$ and let $\eta$ be the unit outward co-normal along $\partial\Sigma$ . The second variation formula says that for any smooth function $\psi$ on $\Sigma$ we have

	$\displaystyle 0$	$\displaystyle\leq\int_{\Sigma}\|\nabla_{\Sigma}\psi^{2}\|-(\|A_{\Sigma}\|^{2}+\operatorname{Ric}(\nu,\nu))\psi^{2}\,dv-\int_{\partial\Sigma}A_{\partial M}(\nu,\nu)\psi^{2}\,da$
		$\displaystyle=-\int_{\Sigma}\psi J_{\Sigma}\psi\,dv+\int_{\partial\Sigma}\psi\frac{\partial\psi}{\partial\eta}-A_{\partial M}(\nu,\nu)\psi^{2}\,da,$

where $J_{\Sigma}=\Delta_{\Sigma}+|A_{\Sigma}|^{2}+\operatorname{Ric}(\nu,\nu)$ is the Jacobi operator. Let $f>0$ be a first eigenfunction so that

\begin{cases}J_{\Sigma}f+\lambda f=0,\\ \partial_{\eta}f=A_{\partial M}(\nu,\nu)f,\end{cases}

with $\lambda\geq 0$ . We are going to consider a weighted length functional with weight $f$ as in Shen-Ye [17].

More precisely, given a curve $c$ in $\Sigma$ , define

L(c)=\int_{0}^{\ell}f|\dot{c}|\,dt.

We now restrict the calculations in $\Sigma$ and follow [17] to compute the first and second variation of $L$ . Assume that $c(t)$ is a unit speed curve in $\Sigma$ . Let $c(s,t)$ be a variation with $c(0,t)=c(t)$ . We compute

\displaystyle\frac{\partial L}{\partial s}=\int_{0}^{\ell}\frac{\partial}{\partial s}(f\langle\dot{c},\dot{c}\rangle^{1/2})\,dt=\int_{0}^{\ell}\langle\nabla f,\frac{\partial c}{\partial s}\rangle|\dot{c}|+f\frac{\langle\nabla_{\partial/\partial s}\dot{c},\dot{c}\rangle}{|\dot{c}|}\,dt.

Therefore setting $V(t)=\frac{\partial c}{\partial s}(0,t)$ we get the first variation formula

\frac{\partial L}{\partial s}\bigg{|}_{s=0}=\int_{0}^{\ell}\langle\nabla f,V\rangle+f\langle\nabla_{\dot{c}}V,\dot{c}\rangle\,dt.

Assuming $c$ is a critical point among curves with endpoints constrained to lie in $\partial\Sigma$ , we can test this against $V$ with $V(0)\in T_{c(0)}\partial\Sigma$ and $V(\ell)\in T_{c(\ell)}\partial\Sigma$ to get

	$\displaystyle 0$	$\displaystyle=\int_{0}^{\ell}\langle\nabla f,V\rangle+f\frac{\partial}{\partial t}\langle V,\dot{c}\rangle-f\langle V,\nabla_{\dot{c}}\dot{c}\rangle\,dt$
		$\displaystyle=\int_{0}^{\ell}\langle\nabla f,V\rangle+\frac{d}{dt}\left[f\langle V,\dot{c}\rangle\right]-\langle\nabla f,\dot{c}\rangle\langle V,\dot{c}\rangle-f\langle V,\nabla_{\dot{c}}\dot{c}\rangle\,dt$
		$\displaystyle=f\langle V,\dot{c}\rangle\bigg{\|}_{0}^{\ell}+\int_{0}^{\ell}\langle\nabla f,V\rangle-\langle\nabla f,\dot{c}\rangle\langle V,\dot{c}\rangle-f\langle V,\nabla_{\dot{c}}\dot{c}\rangle\,dt$
		$\displaystyle=f\langle V,\dot{c}\rangle\bigg{\|}_{0}^{\ell}+\int_{0}^{\ell}\langle(\nabla f)^{\perp},V\rangle-f\langle V,\nabla_{\dot{c}}\dot{c}\rangle\,dt.$

Here $(\nabla f)^{\perp}$ is the part of $\nabla f$ orthogonal to $\dot{c}$ . Therefore $c$ satisfies the weighted geodesic equation

\nabla_{\dot{c}}{\dot{c}}=f^{-1}(\nabla f)^{\perp},

and $c$ meets $\partial\Sigma$ orthogonally at both endpoints. Next we compute the 2nd variation to get

	$\displaystyle\frac{\partial^{2}L}{\partial s^{2}}\bigg{\|}_{s=0}$	$\displaystyle=\int_{0}^{\ell}\langle\nabla_{\frac{\partial c}{\partial s}}\nabla f,V\rangle+\langle\nabla f,\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s}\rangle+2\langle\nabla f,V\rangle\langle\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial t},\frac{\partial c}{\partial t}\rangle+f\frac{\partial}{\partial s}\langle\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial t},\frac{\partial c}{\partial t}\rangle-f\langle\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial t},\frac{\partial c}{\partial t}\rangle^{2}\,dt$
		$\displaystyle=\int_{0}^{\ell}\operatorname{Hess}^{\Sigma}_{f}(V,V)+\langle\nabla f,\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s}\rangle+2\langle\nabla f,V\rangle\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle+f\frac{\partial}{\partial s}\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle-f\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle^{2}\,dt$
		$\displaystyle=\int_{0}^{\ell}\operatorname{Hess}^{\Sigma}_{f}(V,V)+\langle\nabla f,\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s}\rangle+f\langle\nabla_{\frac{\partial c}{\partial s}}\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle\,+f\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial t}\rangle\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}2\langle\nabla f,V\rangle\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle-f\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle^{2}\,dt$
		$\displaystyle=\int_{0}^{\ell}\operatorname{Hess}^{\Sigma}_{f}(V,V)+\langle\nabla f,\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s}\rangle+f\langle\nabla_{\frac{\partial c}{\partial t}}\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle-fR^{\Sigma}(\frac{\partial c}{\partial s},\frac{\partial c}{\partial t},\frac{\partial c}{\partial s},\frac{\partial c}{\partial t})\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}f\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s}\rangle+2\langle\nabla f,V\rangle\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle-f\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle^{2}\,dt$
		$\displaystyle=\int_{0}^{\ell}\operatorname{Hess}^{\Sigma}_{f}(V,V)+\langle\nabla f,\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s}\rangle+f\frac{\partial}{\partial t}\langle\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle-f\langle\nabla_{\frac{\partial c}{\partial s}}\frac{\partial c}{\partial s},\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial t}\rangle-fR^{\Sigma}(\frac{\partial c}{\partial s},\frac{\partial c}{\partial t},\frac{\partial c}{\partial s},\frac{\partial c}{\partial t})\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}f\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s}\rangle+2\langle\nabla f,V\rangle\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle-f\langle\nabla_{\frac{\partial c}{\partial t}}\frac{\partial c}{\partial s},\frac{\partial c}{\partial t}\rangle^{2}\,dt.$

If $c$ minimizes $L$ then this will be non-negative for all admissible variations.

Now assume for contradiction that $\Sigma$ has two distinct boundary components $\Sigma_{1}$ and $\Sigma_{2}$ . Let $c$ be a unit speed curve which minimizes $L$ over all curves connecting $\Sigma_{1}$ to $\Sigma_{2}$ . We select $e_{1}=\dot{c}$ , $e_{2}$ , $\ldots$ , $e_{n}$ to be an orthonormal frame along $c$ . Then

\langle\nabla_{\dot{c}}e_{j},\dot{c}\rangle=-\langle e_{j},\nabla_{\dot{c}}\dot{c}\rangle=-\langle e_{j},f^{-1}\nabla f\rangle

for $j=2,\ldots,n$ . Actually, we can further select $e_{2},\ldots,e_{n}$ to be parallel in the normal bundle of $c$ so that $\nabla_{\dot{c}}e_{j}=-\langle e_{j},f^{-1}\nabla f\rangle\dot{c}$ .

We select variations $c_{j}(s,t)$ with $V_{j}=e_{j}$ and $c_{j}(s,0)\in\Sigma_{1}$ and $c_{j}(s,\ell)\in\Sigma_{2}$ for $j=2,\ldots,n$ . Plugging these into the second variation formula and summing over $j$ we get

	$\displaystyle 0$	$\displaystyle\leq\int_{0}^{\ell}\Delta_{\Sigma}f-\operatorname{Hess}^{\Sigma}_{f}(\dot{c},\dot{c})+\sum_{j=2}^{n}\langle\nabla f,\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s}\rangle+f\sum_{j=2}^{n}\frac{\partial}{\partial t}\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle-f\operatorname{Ric}^{\Sigma}(\dot{c},\dot{c})\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}-f\sum_{j=2}^{n}\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\nabla_{\dot{c}}\dot{c}\rangle+2\sum_{j=2}^{n}\langle\nabla f,e_{j}\rangle\langle\nabla_{\dot{c}}e_{j},\dot{c}\rangle-f\sum_{j=2}^{n}\langle\nabla_{\dot{c}}e_{j},\dot{c}\rangle^{2}+f\sum_{j=2}^{n}\langle\nabla_{\dot{c}}e_{j},\nabla_{\dot{c}}e_{j}\rangle\,dt$
		$\displaystyle=\int_{0}^{\ell}\Delta_{\Sigma}f-\operatorname{Hess}^{\Sigma}_{f}(\dot{c},\dot{c})+\sum_{j=2}^{n}\langle\nabla f,\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s}\rangle+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]-\sum_{j=2}^{n}\langle\nabla f,\dot{c}\rangle\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}-f\sum_{j=2}^{n}\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},f^{-1}(\nabla f)^{\perp}\rangle-2\sum_{j=2}^{n}\langle\nabla f,e_{j}\rangle\langle e_{j},f^{-1}\nabla f\rangle-f\sum_{j=2}^{n}\langle e_{j},f^{-1}\nabla f\rangle^{2}\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}-f\operatorname{Ric}^{\Sigma}(\dot{c},\dot{c})+\sum_{j=2}^{n}f^{-1}\langle\nabla f,e_{j}\rangle^{2}\,dt$
		$\displaystyle=\int_{0}^{\ell}\Delta_{\Sigma}f-\operatorname{Hess}^{\Sigma}_{f}(\dot{c},\dot{c})+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]-2f^{-1}\|(\nabla f)^{\perp}\|^{2}-f\operatorname{Ric}^{\Sigma}(\dot{c},\dot{c})\,dt.$

Now observe that

\operatorname{Hess}_{f}^{\Sigma}(\dot{c},\dot{c})=\langle\nabla_{\dot{c}}\nabla f,\dot{c}\rangle=\frac{d^{2}f}{dt^{2}}-\langle\nabla f,\nabla_{\dot{c}}\dot{c}\rangle=\frac{d^{2}f}{dt^{2}}-f^{-1}|(\nabla f)^{\perp}|^{2}.

Hence we obtain

	$\displaystyle 0$	$\displaystyle\leq\int_{0}^{\ell}-f\|A_{\Sigma}\|^{2}-f\operatorname{Ric}(\nu,\nu)-f\operatorname{Ric}^{\Sigma}(\dot{c},\dot{c})-\frac{d^{2}f}{dt^{2}}+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]-f^{-1}\|(\nabla f)^{\perp}\|^{2}\,dt$
		$\displaystyle=-\int_{0}^{\ell}f\left[\operatorname{Ric}(\nu,\nu)+\operatorname{Ric}(\dot{c},\dot{c})-R(\nu,\dot{c},\nu,\dot{c})\right]\,dt-\int_{0}^{\ell}\frac{d^{2}f}{dt^{2}}\,dt-\int_{0}^{\ell}f^{-1}\|(\nabla f)^{\perp}\|^{2}\,dt$
		$\displaystyle\qquad\quad-\int_{0}^{\ell}f\left[\|A_{\Sigma}\|^{2}+\sum_{j=2}^{n}(A_{\Sigma}(\dot{c},\dot{c})A_{\Sigma}(\dot{e}_{j},e_{j})-A_{\Sigma}(\dot{c},e_{j})^{2})\right]\,dt$
		$\displaystyle\qquad\quad-f(c(\ell))H_{\Sigma_{2}}(c(\ell))-f(c(0))H_{\Sigma_{1}}(c(0))\phantom{\int}$
		$\displaystyle<-\partial_{\eta}f(c(\ell))-\partial_{\eta}f(c(0))-f(c(\ell))H_{\Sigma_{2}}(c(\ell))-f(c(0))H_{\Sigma_{1}}(c(0)).\phantom{\int}$

Here we used the fact that $\operatorname{Ric}(\nu,\nu)+\operatorname{Ric}(\dot{c},\dot{c})-R(\nu,\dot{c},\nu,\dot{c})>0$ by the assumption on the bi-Ricci curvature, and the fact that

|A_{\Sigma}|^{2}+\sum_{j=2}^{n}(A_{\Sigma}(\dot{c},\dot{c})A_{\Sigma}(e_{j},e_{j})-A_{\Sigma}(\dot{c},e_{j})^{2})\geq 0

since $\Sigma$ is minimal; see [17, Equation 15]. Finally, it remains to note that

\partial_{\eta}f=A_{\partial M}(\nu,\nu)f=-fH_{\partial\Sigma}

since $\partial M$ is minimal and $\partial\Sigma$ meets $\partial M$ orthogonally. Thus the final term in the previous chain of inequalities is equal to 0 and we get our contradiction. ∎

Next, we prove a Frankel type theorem for manifolds with positive conformal Ricci curvature.

Theorem 22.

Let $\Sigma$ be a $3$ -dimensional compact Riemannian manifold with boundary. Assume $\Sigma$ admits a positive function $f$ which satisfies

\operatorname{Ric}(v,v)-f^{-1}\Delta f+\frac{1}{2}|\nabla\ln f|^{2}\geq 0

for all unit vectors $v$ . Moreover, suppose that $\partial_{\eta}f=-fH_{\partial\Sigma}$ where $\eta$ is the unit outward normal to $\partial\Sigma$ . Then $\partial\Sigma$ is connected.

Proof.

The argument is similar to the previous one but with a slight improvement coming from a modified choice of the variations. Assume for contradiction that $\partial\Sigma$ has two distinct connected components $\Sigma_{1}$ and $\Sigma_{2}$ . We let $c$ be a unit speed curve which minimizes the $f$ -weighted length from $\Sigma_{1}$ to $\Sigma_{2}$ . Again we let $e_{1}=\dot{c},e_{2},\ldots,e_{n}$ be an orthonormal frame along $c$ such that $e_{2},\ldots,e_{n}$ are parallel in the normal bundle of $c$ . However, this time we choose variations $c_{j}(s,t)$ with $V_{j}=f^{-1/2}e_{j}$ and $c_{j}(s,0)\in\Sigma_{1}$ and $c_{j}(s,\ell)\in\Sigma_{2}$ for $j=2,\ldots,n$ .

Applying the second variation formula to each $c_{j}$ and then summing over $j$ , we deduce that

	$\displaystyle 0$	$\displaystyle\leq\int_{0}^{\ell}f^{-1}\Delta f-f^{-1}\operatorname{Hess}_{f}(\dot{c},\dot{c})+\sum_{j=2}^{n}\langle\nabla f,\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s}\rangle+f\sum_{j=2}^{n}\frac{\partial}{\partial t}\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle-\operatorname{Ric}(\dot{c},\dot{c})\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}-f\sum_{j=2}^{n}\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\nabla_{\dot{c}}\dot{c}\rangle+2\sum_{j=2}^{n}\langle\nabla f,f^{-1/2}e_{j}\rangle\langle\nabla_{\dot{c}}(f^{-1/2}e_{j}),\dot{c}\rangle-f\sum_{j=2}^{n}\langle\nabla_{\dot{c}}(f^{-1/2}e_{j}),\dot{c}\rangle^{2}\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}f\sum_{j=2}^{n}\langle\nabla_{\dot{c}}(f^{-1/2}e_{j}),\nabla_{\dot{c}}(f^{-1/2}e_{j})\rangle\,dt.$

This simplifies to give

	$\displaystyle 0$	$\displaystyle\leq\int_{0}^{\ell}f^{-1}\Delta f-f^{-1}\operatorname{Hess}_{f}(\dot{c},\dot{c})+\sum_{j=2}^{n}\langle\nabla f,\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s}\rangle+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]-\sum_{j=2}^{n}\langle\nabla f,\dot{c}\rangle\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}-f\sum_{j=2}^{n}\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},f^{-1}(\nabla f)^{\perp}\rangle-2f^{-2}\sum_{j=2}^{n}\langle\nabla f,e_{j}\rangle\langle e_{j},\nabla f\rangle-f^{-2}\sum_{j=2}^{n}\langle e_{j},\nabla f\rangle^{2}\,dt$
		$\displaystyle\qquad\quad+\int_{0}^{\ell}-\operatorname{Ric}(\dot{c},\dot{c})+f^{-2}\sum_{j=2}^{n}\langle\nabla f,e_{j}\rangle^{2}+\frac{n-1}{4}\langle\dot{c},\nabla\ln f\rangle^{2}\,dt$
		$\displaystyle=\int_{0}^{\ell}f^{-1}\Delta f-f^{-1}\operatorname{Hess}_{f}(\dot{c},\dot{c})+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]-2f^{-2}\|(\nabla f)^{\perp}\|^{2}-\operatorname{Ric}(\dot{c},\dot{c})+\frac{n-1}{4}\langle\dot{c},\nabla\ln f\rangle^{2}\,dt.$

As before, we have

\operatorname{Hess}_{f}(\dot{c},\dot{c})=\langle\nabla_{\dot{c}}\nabla f,\dot{c}\rangle=\frac{d^{2}f}{dt^{2}}-\langle\nabla f,\nabla_{\dot{c}}\dot{c}\rangle=\frac{d^{2}f}{dt^{2}}-f^{-1}|(\nabla f)^{\perp}|^{2}.

Therefore, combined with $\frac{n-1}{4}\langle\dot{c},\nabla\ln f\rangle^{2}=\frac{1}{2}\langle\dot{c},\nabla\ln f\rangle^{2}\leq\frac{1}{2}|\nabla\ln f|^{2}$ , we obtain

\displaystyle 0

\displaystyle\leq\int_{0}^{\ell}f^{-1}\Delta f-f^{-1}\frac{d^{2}f}{dt^{2}}+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]-f^{-2}|(\nabla f)^{\perp}|^{2}-\operatorname{Ric}(\dot{c},\dot{c})+\frac{1}{2}|\nabla\ln f|^{2}\,dt.

Next, note that

f^{-1}\frac{d^{2}f}{dt^{2}}=\frac{d^{2}}{dt^{2}}(\ln f)+\left[\frac{d(\ln f)}{dt}\right]^{2}=\frac{d^{2}}{dt^{2}}(\ln f)+f^{-2}\langle\nabla f,\dot{c}\rangle^{2}.

It follows that

	$\displaystyle 0$	$\displaystyle\leq\int_{0}^{\ell}f^{-1}\Delta f-f^{-2}\langle\nabla f,\dot{c}\rangle^{2}-f^{-2}\|(\nabla f)^{\perp}\|^{2}+\frac{1}{2}\|\nabla\ln f\|^{2}-\operatorname{Ric}(\dot{c},\dot{c})-\frac{d^{2}}{dt^{2}}(\ln f)+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]\,dt$
		$\displaystyle=\int_{0}^{\ell}f^{-1}\Delta f-\frac{1}{2}\|\nabla\ln f\|^{2}-\operatorname{Ric}(\dot{c},\dot{c})-\frac{d^{2}}{dt^{2}}(\ln f)+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]\,dt$
		$\displaystyle<\int_{0}^{\ell}-\frac{d^{2}}{dt^{2}}(\ln f)+\sum_{j=2}^{n}\frac{\partial}{\partial t}\left[f\langle\nabla_{\frac{\partial c_{j}}{\partial s}}\frac{\partial c_{j}}{\partial s},\dot{c}\rangle\right]\,dt.$

Here we used the first assumption of the theorem to get the final inequality. Now we can apply the fundamental theorem of calculus to get

	$\displaystyle 0$	$\displaystyle<-\partial_{\eta}(\ln f)(c(\ell))-\partial_{\eta}(\ln f)(c(0))-H_{\Sigma_{2}}(c(\ell))-H_{\Sigma_{1}}(c(0))$
		$\displaystyle=-\frac{\partial_{\eta}f}{f}(c(\ell))-\frac{\partial_{\eta}f}{f}(c(0))-H_{\Sigma_{2}}(c(\ell))-H_{\Sigma_{1}}(c(0)).$

The second assumption of the theorem implies that the previous line is equal to 0, and again we’ve reached a contradiction. ∎

4. Proof of the Main Theorem

In this section, we prove our main result Theorem 4. We will proceed case by case based on the value of $m$ .

4.1. The Ricci Curvature Case

Note that the $1$ -intermediate curvature is just the Ricci curvature. Therefore, the $m=1$ case of Conjecture 3 is a well-known corollary of the Bonnet-Myers theorem. In this case, we do not need any restriction on the ambient dimension $n$ .

Theorem 23.

Let $M^{n}$ be a closed manifold. Assume that the universal cover $\overline{M}$ of $M$ satisfies $H_{n}(\overline{M},\mathbb{Z})=0$ . Then $M$ does not admit a metric of positive Ricci curvature.

Proof.

We prove the contrapositive. Assume that $M$ admits a metric of positive Ricci curvature. Let $\overline{M}$ be the universal cover of $M$ . By the Bonnet-Myers theorem, $\overline{M}$ is a closed, orientable $n$ -dimensional manifold. Therefore $H_{n}(\overline{M},\mathbb{Z})\neq 0$ . ∎

4.2. The Scalar Curvature Case

We now show that the $m=n-1$ case of Conjecture 3 is equivalent to the $K(\pi,1)$ conjecture. Since the $(n-1)$ -intermediate curvature is equivalent to scalar curvature, it suffices to prove the following simple fact.

Theorem 24.

Let $M^{n}$ be a closed manifold. Then the universal cover $\overline{M}$ of $M$ satisfies

(3)

H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=\ldots=H_{2}(\overline{M},\mathbb{Z})=0

if and only if $M$ is aspherical.

Proof.

First suppose that $M$ is aspherical. Then $\overline{M}$ is contractible and so it is immediate that (3) holds. Conversely, suppose that $\overline{M}$ satisfies (3). Since $\overline{M}$ is an $n$ -dimensional manifold, (3) implies that $H_{k}(\overline{M},\mathbb{Z})=0$ for all $k\geq 2$ . As $\overline{M}$ is simply connected, it then follows from the Hurewicz theorem [11, Theorem 4.32] that $\pi_{k}(\overline{M})=0$ for all $k\geq 1$ . The Whitehead theorem [11, Theorem 4.5] now implies that $\overline{M}$ is contractible and so $M$ is aspherical. ∎

Since the $K(\pi,1)$ conjecture is known to be true for $n\in\{3,4,5\}$ , we have the following immediate corollary.

Corollary 25.

Conjecture 3 is true for $n\in\{3,4,5\}$ and $m=n-1$ .

4.3. The Codimension Two Case

In this section, we prove the $m=2$ case of Conjecture 3 for $n\in\{3,4,5,6\}$ . Our argument combines a construction of Chodosh-Li [3] with a diameter bound of Shen-Ye [17, 18].

Theorem 26.

Let $M^{n}$ be a closed manifold of dimension $n\in\{3,4,5,6\}$ , Assume that the universal cover $\overline{M}$ of $M$ satisfies $H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=0$ . Then $M$ does not admit a metric of $2$ -positive intermediate curvature.

Proof.

Let $M$ be as in the statement of the theorem. Assume for the sake of contradiction that $M$ admits a metric with positive 2-intermediate curvature. By scaling, we can suppose that the 2-intermediate curvature of $M$ is at least $2$ . For a fixed large constant $L>0$ , we can apply Proposition 13 to find a geodesic line $\sigma$ in $\overline{M}$ and a closed manifold $\Lambda^{n-2}$ embedded in $\overline{M}$ such that $\Lambda$ is linked with $\sigma$ and $d(\sigma,\Lambda)\geq L$ . Now let $\Sigma_{1}$ be the area minimizing minimal hypersurface in $\overline{M}$ with $\partial\Sigma_{1}=\Lambda$ . Then $\Sigma_{1}$ must intersect $\sigma$ and so there is a point $p\in\Sigma_{1}$ with $d(p,\partial\Sigma_{1})\geq L$ . For $n\in\{3,4,5\}$ , this contradicts the Shen-Ye [17] diameter estimate for stable minimal hypersurfaces in manifolds with positive 2-intermediate curvature (Theorem 17) provided $L$ is chosen large enough.

When $n=6$ , we need to argue more carefully to get the diameter bound. After establishing the diameter bound, one then gets a contradiction in the same way. Choose $\tau<1$ close enough to $1$ so that $(1-\tau)|\operatorname{Ric}_{M}(v,v)|<1$ for all unit vectors $v$ . Let $\Sigma_{1}$ be constructed as above, and let $f>0$ be the first eigenfunction of the stability operator so that

\Delta_{\Sigma_{1}}f+|A_{\Sigma_{1}}|^{2}f+\operatorname{Ric}_{\overline{M}}(\nu,\nu)f\leq 0.

Since $\Sigma_{1}$ is five dimensional, Theorem 20 says that if

\operatorname{Ric}_{\Sigma_{1}}(v,v)-\tau f^{-1}\Delta_{\Sigma_{1}}f+\left[\tau-(1+\varepsilon)\tau^{2}\right]|\nabla_{\Sigma_{1}}\ln f|^{2}\geq\kappa>0

for all unit vectors $v\in T_{p}\Sigma_{1}$ , then the diameter of $\Sigma_{1}$ is bounded above in terms of $n$ , $\kappa$ , and $\varepsilon$ . Importantly, we can choose $\varepsilon=\varepsilon(\tau)$ close enough to 0 that $\tau-(1+\varepsilon)\tau^{2}\geq 0$ .

Thus it is enough to show that

\operatorname{Ric}_{\Sigma_{1}}(v,v)-\tau f^{-1}\Delta_{\Sigma_{1}}f\geq\kappa>0

for all unit vectors $v\in T_{p}\Sigma_{1}$ . Let $e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}$ be an orthonormal basis for $T_{p}\overline{M}$ for which $e_{2},e_{3},e_{4},e_{5},e_{6}$ is an orthonormal basis for $T_{p}\Sigma_{1}$ . We compute

	$\displaystyle\operatorname{Ric}_{\Sigma_{1}}$	$\displaystyle(e_{2},e_{2})-\tau f^{-1}\Delta_{\Sigma_{1}}f\phantom{\int}$
		$\displaystyle\geq\operatorname{Ric}_{\Sigma_{1}}(e_{2},e_{2})+\tau\|A_{\Sigma_{1}}\|^{2}+\tau\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})$
		$\displaystyle=\operatorname{Ric}_{\overline{M}}(e_{2},e_{2})+\tau\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})-R_{\overline{M}}(e_{1},e_{2},e_{1},e_{2})+\tau\|A_{\Sigma_{1}}\|^{2}+\sum_{j=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{jj}-(A^{\Sigma_{1}}_{2j})^{2}\right]$
		$\displaystyle\geq 2+(\tau-1)\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})+\tau\|A_{\Sigma_{1}}\|^{2}+\sum_{j=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{jj}-(A^{\Sigma_{1}}_{2j})^{2}\right]$
		$\displaystyle\geq 1+\tau\|A_{\Sigma_{1}}\|^{2}+\sum_{j=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{jj}-(A^{\Sigma_{1}}_{2j})^{2}\right].$

It remains to show that the second fundamental form terms give a non-negative contribution. We will drop the superscript $\Sigma_{1}$ in what follows. Since $\Sigma_{1}$ is minimal, we have

\displaystyle\sum_{j=3}^{6}\left[A_{22}A_{jj}-A_{2j}^{2}\right]

\displaystyle=-\sum_{i=3}^{6}\sum_{j=3}^{6}A_{ii}A_{jj}-\sum_{j=3}^{6}A_{2j}^{2}=-2\sum_{3\leq i<j\leq 6}A_{ii}A_{jj}-\sum_{j=3}^{6}A_{jj}^{2}-\sum_{j=3}^{6}A_{2j}^{2}.

Also we have

	$\displaystyle\|A\|^{2}$	$\displaystyle\geq A_{22}^{2}+\sum_{j=3}^{6}A_{jj}^{2}+2\sum_{j=3}^{6}A_{2j}^{2}$
		$\displaystyle=\left(\sum_{j=3}^{6}A_{jj}\right)^{2}+\sum_{j=3}^{6}A_{jj}^{2}+2\sum_{j=3}^{6}A_{2j}^{2}$
		$\displaystyle=2\sum_{3\leq i<j\leq 6}A_{ii}A_{jj}+2\sum_{j=3}^{6}A_{jj}^{2}+2\sum_{j=3}^{6}A_{2j}^{2}.$

So assuming $\tau\geq 1/2$ we get

(4)

\displaystyle\tau|A|^{2}+\sum_{j=3}^{6}\left[A_{22}A_{jj}-A_{2j}^{2}\right]

\displaystyle\geq(2\tau-2)\sum_{3\leq i<j\leq 6}A_{ii}A_{jj}+(2\tau-1)\sum_{j=3}^{6}A_{jj}^{2}.

Finally, note that

\displaystyle\sum_{j=3}^{6}A_{jj}^{2}=\frac{1}{3}\sum_{3\leq i<j\leq 6}(A_{ii}^{2}+A_{jj}^{2})\geq\frac{2}{3}\sum_{3\leq i<j\leq 6}|A_{ii}A_{jj}|.

Combining this with (4), we obtain

\tau|A|^{2}+\sum_{j=3}^{6}\left[A_{22}A_{jj}-A_{2j}^{2}\right]\geq 0

as long as

\frac{2-2\tau}{2\tau-1}<\frac{2}{3},

which we can always ensure by choosing $\tau$ close enough to 1. This completes the proof.

∎

4.4. The Case $n=5$ and $m=3$

Next we show that Conjecture 3 is true for $n=5$ and $m=3$ . This will complete the proof of Theorem 4 for $n\in\{3,4,5\}$ .

Theorem 27.

Let $M^{5}$ be a closed manifold. Assume that the universal cover $\overline{M}$ of $M$ satisfies

H_{5}(\overline{M},\mathbb{Z})=H_{4}(\overline{M},\mathbb{Z})=H_{3}(\overline{M},\mathbb{Z})=0.

Then $M$ does not admit a metric with positive 3-intermediate curvature.

Let $M$ be as in the statement of the theorem. Assume for the sake of contradiction that $M$ admits a metric with positive $3$ -intermediate curvature. By scaling, we can suppose the 3-intermediate curvature of $M$ is at least $1$ . For a fixed large $L>0$ , we apply Proposition 13 to find a geodesic line $\sigma$ in $\overline{M}$ and a closed manifold $\Lambda^{n-2}$ embedded in $\overline{M}$ such that $\Lambda$ is linked with $\sigma$ and $d(\sigma,\Lambda)\geq 2L$ . Let $\Sigma_{1}$ be the area minimizing minimal hypersurface in $\overline{M}$ with $\partial\Sigma_{1}=\Lambda$ . Let $\eta$ be the unit normal to $\Sigma_{1}$ . Since $\Sigma_{1}$ is area minimizing, there exists a function $u>0$ on $\Sigma_{1}$ which satisfies $\Delta_{\Sigma_{1}}u+(|A_{\Sigma_{1}}|^{2}+\operatorname{Ric}_{\overline{M}}(\eta,\eta))u\leq 0$ .

Next, we argue as in Chodosh-Li [3] to construct a suitable function $h$ on $\Sigma_{1}$ .Define

\rho_{0}(x)=d_{\overline{M}}(x,\sigma),\quad x\in\overline{M}.

Then let $\rho_{1}$ be a smooth approximation to $\rho_{0}$ which satisfies $\|\nabla\rho_{1}(x)\|\leq 2$ for all $x\in\overline{M}$ , $\rho_{1}<1$ on $\sigma$ , and $\rho_{1}>L+4\pi+1$ on $\partial\Sigma_{1}=\Lambda$ . Let $\rho_{2}$ be the restriction of $\rho_{1}$ to $\Sigma_{1}$ . Now choose $\varepsilon>0$ sufficiently small so that $L-\varepsilon$ , $L+2\pi(1+\varepsilon)+\varepsilon^{2}$ , and $L+4\pi+\varepsilon$ are all regular values of $\rho_{2}$ . We can further ensure that $\Sigma_{1}\cap B_{\overline{M}}(\sigma,L/4)\subset\{\rho_{2}\leq L-\varepsilon\}$ . More precisely, assume $L$ is chosen so that $L/4<(L-\epsilon-1)/2$ . Then for $q\in\Sigma_{1}\cap B_{\overline{M}}(\sigma,L/4)$ we have

\rho_{2}(q)=\rho_{1}(q)\leq\sup_{\sigma}\rho_{1}+(L/4)\cdot\|\nabla\rho_{1}\|_{L^{\infty}}\leq L-\epsilon,

so $\Sigma_{1}\cap B_{\overline{M}}(\sigma,L/4)\subset\{\rho_{2}\leq L-\varepsilon\}$ . Now define

\rho(x)=\frac{\rho_{2}(x)-L-2\pi}{2\pi+\varepsilon},\quad x\in\Sigma_{1}.

Then we have $\|\nabla_{\Sigma_{1}}\rho(x)\|\leq\frac{1}{\pi}$ for all $x\in\Sigma_{1}$ . We define

	$\displaystyle\tilde{\Sigma}_{1}=\{-1\leq\rho\leq 1\},$
	$\displaystyle\partial_{+}\tilde{\Sigma}_{1}=\{\rho=-1\},$
	$\displaystyle\partial_{-}\tilde{\Sigma}_{1}=\{\rho=1\},$
	$\displaystyle\Omega_{0}=\{-1\leq\rho<\varepsilon\}.$

By construction, $\partial_{\pm}\tilde{\Sigma}_{1}$ and $\partial\Omega_{0}$ are all smooth hypersurfaces in $\tilde{\Sigma}_{1}$ , and $\Omega_{0}$ contains $\partial_{+}\tilde{\Sigma}_{1}$ , and $\partial\Sigma_{1}\cap\tilde{\Sigma}_{1}=\emptyset$ . Finally, we define

h(x)=-\tan\left(\frac{\pi\rho(x)}{2}\right).

Then $h$ is a smooth function on the interior of $\tilde{\Sigma}_{1}$ which satisfies $h(x)\to\pm\infty$ as $x\to\partial_{\pm}\tilde{\Sigma}_{1}$ . Moreover, we have

\|\nabla_{\Sigma_{1}}h(x)\|=\frac{\pi}{2}\sec^{2}\left(\frac{\pi\rho(x)}{2}\right)\|\nabla_{\Sigma_{1}}\rho(x)\|\leq\frac{1}{2}\left(1+\tan^{2}\left(\frac{\pi\rho(x)}{2}\right)\right)=\frac{1}{2}\left(1+h^{2}\right)

for all $x\in\tilde{\Sigma}_{1}$ . It follows that

(5)

1+h^{2}-2\|\nabla_{\Sigma_{1}}h\|\geq 0

on $\tilde{\Sigma}_{1}$ . Proposition 15 implies that there exists a minimizer $\Omega\subset\tilde{\Sigma}_{1}$ of the $\mu$ -bubble functional

\mathcal{A}(\Omega)=\int_{\partial\Omega}u\,d\mathcal{H}^{3}-\int_{\tilde{\Sigma}_{1}}(\chi_{\Omega}-\chi_{\Omega_{0}})hu\,d\mathcal{H}^{4}

such that $\Omega\operatorname{\Delta}\Omega_{0}$ is compactly contained in the interior of $\tilde{\Sigma}_{1}$ .

Let $\Sigma_{2}$ be a connected component of $\partial\Omega$ , and consider the slicing $\Sigma_{2}\subset\Sigma_{1}\subset\overline{M}$ together with the functions $u$ and $h$ .

We are going to show that $\operatorname{diam}(\Sigma_{2})$ is bounded from above by a universal constant that does not depend on $L$ . Let $\nu$ be the unit normal vector to $\Sigma_{2}$ in $\Sigma_{1}$ pointing out of $\Omega$ . According to Proposition 16, the mean curvature of $\Sigma_{2}$ satisfies

H_{\Sigma_{2}}=h-u^{-1}\langle\nabla_{\Sigma_{1}}u,\nu\rangle.

In the next proposition, we re-arrange the second variation formula into a more convenient form.

Proposition 28.

For any $\psi\in C^{1}(\Sigma_{2})$ we have

\displaystyle 0\leq\int_{\Sigma_{2}}|\nabla_{\Sigma_{2}}\psi|^{2}u-\left(|A_{\Sigma_{1}}|^{2}+\operatorname{Ric}_{\overline{M}}(\eta,\eta)+|A_{\Sigma_{2}}|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)-\frac{1}{2}H_{\Sigma_{2}}^{2}-\frac{1}{2}+\frac{\Delta_{\Sigma_{2}}u}{u}\right)\psi^{2}u

Proof.

The second variation formula (Proposition 16) and $H_{\Sigma_{2}}=h-u^{-1}\langle\nabla_{\Sigma_{1}}u,\nu\rangle$ give that

0\leq\int_{\Sigma_{2}}|\nabla_{\Sigma_{2}}\psi|^{2}u-(|A_{\Sigma_{2}}|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu))\psi^{2}u-\langle\nabla_{\Sigma_{1}}h,\nu\rangle\psi^{2}u-h\langle\nabla_{\Sigma_{1}}u,\nu\rangle\psi^{2}+(\Delta_{\Sigma_{1}}u-\Delta_{\Sigma_{2}}u)\psi^{2}.

We now simplify this as in Chodosh-Li [3]. First, observe that

\frac{1}{2}H_{\Sigma_{2}}^{2}\psi^{2}u=\frac{1}{2}u^{-1}\langle\nabla_{\Sigma_{1}}u,\nu\rangle^{2}\psi^{2}-h\langle\nabla_{\Sigma_{1}}u,\nu\rangle\psi^{2}+\frac{1}{2}h^{2}\psi^{2}u.

Using this to eliminate the $h\langle\nabla_{\Sigma_{1}}u,\nu\rangle\psi^{2}$ term in the previous inequality, we see that

	$\displaystyle 0$	$\displaystyle\leq\int_{\Sigma_{2}}\|\nabla_{\Sigma_{2}}\psi\|^{2}u-(\|A_{\Sigma_{2}}\|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)-\frac{1}{2}H_{\Sigma_{2}}^{2})\psi^{2}u+(\Delta_{\Sigma_{1}}u-\Delta_{\Sigma_{2}}u)\psi^{2}$
		$\displaystyle\qquad\qquad-\frac{1}{2}\int_{\Sigma_{2}}\left(\langle u^{-1}\nabla_{\Sigma_{1}}u,\nu\rangle^{2}+h^{2}+2\langle\nabla_{\Sigma_{1}}h,\nu\rangle\right)\psi^{2}u.$

Since $\langle u^{-1}\nabla_{\Sigma_{1}}u,\nu\rangle^{2}\geq 0$ and $1+h^{2}+2\langle\nabla_{\Sigma_{1}}h,\nu\rangle\geq 0$ by (5), it follows that

\displaystyle 0\leq\int_{\Sigma_{2}}|\nabla_{\Sigma_{2}}\psi|^{2}u-\left(|A_{\Sigma_{2}}|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)-\frac{1}{2}H_{{\Sigma_{2}}}^{2}-\frac{1}{2}\right)\psi^{2}u+(\Delta_{\Sigma_{1}}u-\Delta_{\Sigma_{2}}u)\psi^{2}.

Finally, recalling that $\Delta_{\Sigma_{1}}u+(|A_{\Sigma_{1}}|^{2}+\operatorname{Ric}_{\overline{M}}(\eta,\eta))u\leq 0$ , we get

0\leq\int_{\Sigma_{2}}|\nabla_{\Sigma_{2}}\psi|^{2}u-\left(|A_{\Sigma_{1}}|^{2}+\operatorname{Ric}_{\overline{M}}(\eta,\eta)+|A_{\Sigma_{2}}|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)-\frac{1}{2}H_{\Sigma_{2}}^{2}-\frac{1}{2}+\frac{\Delta_{\Sigma_{2}}u}{u}\right)\psi^{2}u,

as needed. ∎

Now we can argue that the diameter of ${\Sigma_{2}}$ is bounded.

Proposition 29.

The diameter $\operatorname{diam}({\Sigma_{2}})$ of $\Sigma_{2}$ is bounded uniformly from above.

Proof.

Note that Proposition 28 implies that there is a function $w>0$ on ${\Sigma_{2}}$ which satisfies

(6)

\operatorname{div}_{\Sigma_{2}}(u\nabla_{\Sigma_{2}}w)\leq-\left(|A_{\Sigma_{2}}|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)+|A_{\Sigma_{1}}|^{2}+\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\frac{1}{2}H_{{\Sigma_{2}}}^{2}-\frac{1}{2}+\frac{\Delta_{\Sigma_{2}}u}{u}\right)wu.

Our strategy is to apply Theorem 18 with $N={\Sigma_{2}}$ and $f=uw$ .

Fix a point $p\in{\Sigma_{2}}$ and let $\{e_{1}=\eta,e_{2}=\nu,e_{3},e_{4},e_{5}\}$ be an orthonormal frame at $p$ such that $\{e_{2},e_{3},e_{4},e_{5}\}$ is an orthonormal basis for ${\Sigma_{1}}$ and $\{e_{3},e_{4},e_{5}\}$ is an orthonormal basis for ${\Sigma_{2}}$ . We compute that

	$\displaystyle\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\frac{\Delta_{\Sigma_{2}}(uw)}{uw}+\frac{1}{2}\|\nabla_{\Sigma_{2}}\log(uw)\|^{2}$
	$\displaystyle\qquad\qquad=\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\frac{\operatorname{div}_{\Sigma_{2}}(u\nabla_{\Sigma_{2}}w)}{uw}-\frac{\operatorname{div}_{\Sigma_{2}}(w\nabla_{\Sigma_{2}}u)}{uw}+\frac{1}{2}\left\|\frac{\nabla_{\Sigma_{2}}u}{u}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}$
	$\displaystyle\qquad\qquad\geq\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\frac{\operatorname{div}_{\Sigma_{2}}(u\nabla_{\Sigma_{2}}w)}{uw}-\frac{\Delta_{\Sigma_{2}}u}{u}.$

Therefore, by inequality (6), we obtain

	$\displaystyle\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\frac{\Delta_{\Sigma_{2}}(uw)}{uw}+\frac{1}{2}\|\nabla_{\Sigma_{2}}\log(uw)\|^{2}$
	$\displaystyle\qquad\qquad\geq\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})+\operatorname{Ric}_{\Sigma_{1}}(e_{2},e_{2})+\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})+\|A_{\Sigma_{1}}\|^{2}+\|A_{\Sigma_{2}}\|^{2}-\frac{1}{2}H_{\Sigma_{2}}^{2}-\frac{1}{2}.$

It remains to get a lower bound on the right hand side.

According to [2, Lemma 3.8], we have

\displaystyle\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})+\operatorname{Ric}_{\Sigma_{1}}(e_{2},e_{2})+\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})=C_{3}(e_{1},e_{2},e_{3})+\mathcal{B}

where

\mathcal{B}=\sum_{p=2}^{3}\sum_{q=p+1}^{5}\big{(}A_{\Sigma_{1}}(e_{p},e_{p})A_{\Sigma_{1}}(e_{q},e_{q})-A_{{\Sigma_{1}}}(e_{p},e_{q})^{2}\big{)}+\sum_{q=4}^{5}\big{(}A_{\Sigma_{2}}(e_{3},e_{3})A_{\Sigma_{2}}(e_{q},e_{q})-A_{\Sigma_{2}}(e_{3},e_{q})^{2}\big{)}.

Now, since ${\Sigma_{1}}$ is minimal, [2, Lemma 3.11] implies that

|A_{\Sigma_{1}}|^{2}+\sum_{p=2}^{3}\sum_{q=p+1}^{5}\big{(}A_{\Sigma_{1}}(e_{p},e_{p})A_{\Sigma_{1}}(e_{q},e_{q})-A_{{\Sigma_{1}}}(e_{p},e_{q})^{2}\big{)}\geq 0.

Moreover, we have

	$\displaystyle\|A_{\Sigma_{2}}\|^{2}-\frac{1}{2}H_{\Sigma_{2}}^{2}+\sum_{q=4}^{5}\big{(}A_{\Sigma_{2}}(e_{3},e_{3})A_{\Sigma_{2}}(e_{q},e_{q})-A_{\Sigma_{2}}(e_{3},e_{q})^{2}\big{)}$
	$\displaystyle\qquad=\frac{1}{2}(A^{\Sigma_{2}}_{33})^{2}+\frac{1}{2}(A^{\Sigma_{2}}_{44})^{2}+\frac{1}{2}(A^{\Sigma_{2}}_{55})^{2}-A^{\Sigma_{2}}_{44}A^{\Sigma_{2}}_{55}+(A^{\Sigma_{2}}_{34})^{2}+(A^{\Sigma_{2}}_{35})^{2}+2(A^{\Sigma_{2}}_{45})^{2}\geq 0.$

Hence we obtain

	$\displaystyle\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\frac{\Delta_{\Sigma_{2}}(uw)}{uw}+\frac{1}{2}\|\nabla_{\Sigma_{2}}\log(uw)\|^{2}$
	$\displaystyle\qquad\qquad\geq C_{3}(e_{1},e_{2},e_{3})+\mathcal{B}+\|A_{\Sigma_{1}}\|^{2}+\|A_{\Sigma_{2}}\|^{2}-\frac{1}{2}H_{\Sigma_{2}}^{2}-\frac{1}{2}$
	$\displaystyle\qquad\qquad\geq C_{3}(e_{1},e_{2},e_{3})-\frac{1}{2}\geq\frac{1}{2}.$

Since $e_{3}$ was an arbitrary unit tangent vector to ${\Sigma_{2}}$ and $\Sigma_{2}$ is 3-dimensional, Shen-Ye’s generalized Bonnet-Myers theorem (Theorem 18) now gives the conclusion. ∎

We can now finish the proof of Theorem 27. Since $\partial{\Sigma_{2}}=0$ and $\operatorname{diam}({\Sigma_{2}})$ is uniformly bounded and $H_{3}(\overline{M},\mathbb{Z})=0$ , it follows from Proposition 14 that ${\Sigma_{2}}$ bounds within an $R$ -neighborhood for some constant $R$ that depends only on $\overline{M}$ . Applying this argument to each component of $\partial\Omega$ , we see that $\partial\Omega$ bounds within its $R$ -neighborhood in $\overline{M}$ . Since $d(\partial\Omega,\sigma)\geq L/4$ and $\partial\Omega$ is linked with $\sigma$ by construction, this is a contradiction provided we select $L>2R$ . This completes the proof.

4.5. The Case $n=6$ and $m=4$

Next we prove the $n=6$ and $m=4$ case of Theorem 4. Our argument combines the slice and dice procedure of Chodosh-Li [3], the diameter bound of Shen-Ye [17], and also the Frankel type theorems we developed in Section 3.

Theorem 30.

Let $M^{6}$ be a closed 6-dimensional manifold. Assume the universal cover $\overline{M}$ of $M$ satisfies $H_{6}(\overline{M},\mathbb{Z})=H_{5}(\overline{M},\mathbb{Z})=H_{4}(\overline{M},\mathbb{Z})=H_{3}(\overline{M},\mathbb{Z})=0$ . Then $M$ does not admit a metric of positive $4$ -intermediate curvature.

For $n=6$ , $m=4$ , let $\Sigma_{2}\subset\Sigma_{1}\subset\overline{M}^{6}$ be constructed as in the previous subsection, which are now of dimension $4$ and $5$ . We consider the following weighted functional for $3$ -dimensional $\Sigma_{3}\subset{\Sigma_{2}}$ :

\mathcal{A}_{2}(\Sigma_{3}):=\int_{\Sigma_{3}}uw.

We will use the $\mathcal{A}_{2}$ -functional to slice ${\Sigma_{2}}$ into a $4$ -manifold $\hat{{\Sigma}}_{2}$ with simple third homology. Firstly, we prove that each boundary component of $\hat{{\Sigma}}_{2}$ has finite diameter. Without loss of generality, we assume the $4$ -intermediate curvature of $M$ is at least $\frac{3}{2}$ .

Lemma 31.

Suppose $\Sigma_{3}$ is a connected (two-sided) stable critical point of $\mathcal{A}_{2}$ . Then $\operatorname{diam}\Sigma_{3}\leq 2\pi$ and $H_{1}(\Sigma_{3})$ is finite.

Proof.

By the first variation formula, we have

H_{\Sigma_{3}}=-\langle\nabla_{{\Sigma_{2}}}\log(uw),\omega\rangle,

where $H_{\Sigma_{3}}$ is the mean curvature of $\Sigma_{3}$ with respect to the unit normal $\omega$ of $\Sigma_{3}$ in ${\Sigma_{2}}$ . As before, let $\{e_{i}\}_{i=1}^{6}$ be an orthonormal basis of $T_{p}\overline{M}$ so that $e_{j}\perp\Sigma_{j}$ for $j=1,2,3$ , i.e. $e_{1}=\eta,e_{2}=\nu,e_{3}=\omega$ .

By the second variation formula for $\mathcal{A}_{2}$ , there exists a positive function $v\in C^{\infty}(\Sigma_{3})$ such that

-v^{-1}\Delta_{\Sigma_{3}}v\geq\|A_{\Sigma_{3}}\|^{2}+\operatorname{Ric}_{{\Sigma_{2}}}(\omega,\omega)-\nabla_{{\Sigma_{2}}}^{2}\log(uw)(\omega,\omega)+\langle\nabla_{\Sigma_{3}}\log(uw),\nabla_{\Sigma_{3}}\log v\rangle.

We let $f=uwv$ , and our aim is to prove

\operatorname{Ric}_{\Sigma_{3}}(e,e)-f^{-1}\Delta_{\Sigma_{3}}f+\frac{1}{2}|\nabla_{\Sigma_{3}}\log f|^{2}\geq\frac{1}{2}>0,

for any unit vector $e\in T\Sigma_{3}$ .

Suppose $f_{2}$ is a smooth function defined on ${\Sigma_{2}}$ . Then we have

(7)

\displaystyle\Delta_{{\Sigma_{2}}}f_{2}=\Delta_{\Sigma_{3}}f_{2}+\nabla_{{\Sigma_{2}}}^{2}f_{2}(\omega,\omega)-\langle\nabla_{{\Sigma_{2}}}f_{2},\omega\rangle H_{\Sigma_{3}}.

By (6), we have

	$\displaystyle(uw)^{-1}$	$\displaystyle\Delta_{{\Sigma_{2}}}(uw)$
	$\displaystyle\leq$	$\displaystyle-\\|A_{{\Sigma_{1}}}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\\|A_{{\Sigma_{2}}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{1}}}(\nu,\nu)+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\langle\nabla_{{\Sigma_{2}}}\log u,\nabla_{{\Sigma_{2}}}\log w\rangle+\frac{1}{2},$
	$\displaystyle=$	$\displaystyle-\\|A_{{\Sigma_{1}}}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\\|A_{{\Sigma_{2}}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{1}}}(\nu,\nu)+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}$
		$\displaystyle+\langle\nabla_{{\Sigma_{2}}}\log u,\omega\rangle\langle\nabla_{{\Sigma_{2}}}\log w,\omega\rangle+\langle\nabla_{\Sigma_{3}}\log u,\nabla_{\Sigma_{3}}\log w\rangle.$

Using (7) and the first variation formula, we can further write

	$\displaystyle(uw)^{-1}\Delta_{\Sigma_{3}}(uw)=$	$\displaystyle(uw)^{-1}\Delta_{{\Sigma_{2}}}(uw)-\nabla_{{\Sigma_{2}}}^{2}\log(uw)(\omega,\omega)$
	$\displaystyle\leq$	$\displaystyle-\nabla_{{\Sigma_{2}}}^{2}\log(uw)(\omega,\omega)+\langle\nabla_{{\Sigma_{2}}}\log u,\omega\rangle\langle\nabla_{{\Sigma_{2}}}\log w,\omega\rangle+\langle\nabla_{\Sigma_{3}}\log u,\nabla_{\Sigma_{3}}\log w\rangle$
		$\displaystyle-\\|A_{{\Sigma_{1}}}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\\|A_{{\Sigma_{2}}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{1}}}(\nu,\nu)+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}.$

We now have that

	$\displaystyle(uwv)^{-1}\Delta_{\Sigma_{3}}(uwv)=$	$\displaystyle(uw)^{-1}\Delta_{\Sigma_{3}}(uw)+v^{-1}\Delta_{\Sigma_{3}}v+2\langle\nabla_{\Sigma_{3}}\log(uw),\nabla_{\Sigma_{3}}\log v\rangle$
	$\displaystyle\leq$	$\displaystyle\langle\nabla_{{\Sigma_{2}}}\log u,\omega\rangle\langle\nabla_{{\Sigma_{2}}}\log w,\omega\rangle+\langle\nabla_{\Sigma_{3}}\log u,\nabla_{\Sigma_{3}}\log w\rangle+\langle\nabla_{{\Sigma_{3}}}\log(uw),\nabla_{{\Sigma_{1}}}\log v\rangle$
		$\displaystyle-\\|A_{{\Sigma_{1}}}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\\|A_{{\Sigma_{2}}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{1}}}(\nu,\nu)-\\|A_{\Sigma_{3}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{2}}}(\omega,\omega)+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}$
	$\displaystyle\leq$	$\displaystyle\frac{1}{2}\left(\langle\nabla_{{\Sigma_{2}}}\log u,\omega\rangle+\langle\nabla_{{\Sigma_{2}}}\log w,\omega\rangle\right)^{2}+\langle\nabla_{\Sigma_{3}}\log u,\nabla_{\Sigma_{3}}\log w\rangle+\langle\nabla_{{\Sigma_{3}}}\log(uw),\nabla_{{\Sigma_{1}}}\log v\rangle$
		$\displaystyle-\\|A_{{\Sigma_{1}}}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\\|A_{{\Sigma_{2}}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{1}}}(\nu,\nu)-\\|A_{\Sigma_{3}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{2}}}(\omega,\omega)+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}$
	$\displaystyle=$	$\displaystyle\langle\nabla_{\Sigma_{3}}\log u,\nabla_{\Sigma_{3}}\log w\rangle+\langle\nabla_{{\Sigma_{3}}}\log(uw),\nabla_{{\Sigma_{1}}}\log v\rangle+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}H^{2}_{\Sigma_{3}}+\frac{1}{2}$
		$\displaystyle-\\|A_{{\Sigma_{1}}}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\\|A_{{\Sigma_{2}}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{1}}}(\nu,\nu)-\\|A_{\Sigma_{3}}\\|^{2}-\operatorname{Ric}_{{\Sigma_{2}}}(\omega,\omega).$

We used the first variation in the last equality.

Summarizing the above computations, we have

	$\displaystyle\operatorname{Ric}_{\Sigma_{3}}(e_{4},e_{4})$	$\displaystyle-(uwv)^{-1}\Delta_{\Sigma_{3}}(uwv)+\frac{1}{2}\|\nabla_{\Sigma_{3}}\log(uwv)\|^{2}$
	$\displaystyle\geq$	$\displaystyle\\|A_{{\Sigma_{1}}}\\|^{2}+\\|A_{{\Sigma_{2}}}\\|^{2}+\\|A_{\Sigma_{3}}\\|^{2}-\frac{1}{2}H_{{\Sigma_{2}}}^{2}-\frac{1}{2}H_{\Sigma_{3}}^{2}-\frac{1}{2}$
		$\displaystyle+\operatorname{Ric}_{\Sigma_{3}}(e_{4},e_{4})+\operatorname{Ric}_{{\Sigma_{2}}}(\omega,\omega)+\operatorname{Ric}_{{\Sigma_{1}}}(\nu,\nu)+\operatorname{Ric}_{\overline{M}}(\eta,\eta)$
	$\displaystyle=$	$\displaystyle\\|A_{{\Sigma_{1}}}\\|^{2}+\\|A_{{\Sigma_{2}}}\\|^{2}+\\|A_{\Sigma_{3}}\\|^{2}-\frac{1}{2}H_{{\Sigma_{2}}}^{2}-\frac{1}{2}H_{\Sigma_{3}}^{2}-\frac{1}{2}$
		$\displaystyle+C_{4}(e_{1},e_{2},e_{3},e_{4})+\sum_{p=2}^{4}\sum_{q=p+1}^{6}\left(A_{{\Sigma_{1}}}(e_{p},e_{p})A_{{\Sigma_{1}}}(e_{q},e_{q})-A_{{\Sigma_{1}}}(e_{p},e_{q})^{2}\right)$
		$\displaystyle+\sum_{p=3}^{4}\sum_{q=p+1}^{6}\left(A_{{\Sigma_{2}}}(e_{p},e_{p})A_{{\Sigma_{2}}}(e_{q},e_{q})-A_{{\Sigma_{2}}}(e_{p},e_{q})^{2}\right)$
		$\displaystyle+\sum_{q=5}^{6}\left(A_{\Sigma_{3}}(e_{4},e_{4})A_{\Sigma_{3}}(e_{q},e_{q})-A_{\Sigma_{3}}(e_{4},e_{q})^{2}\right)$
	$\displaystyle\geq$	$\displaystyle C_{4}(e_{1},e_{2},e_{3},e_{4})-\frac{1}{2}\geq 1.$

To get the last line above, we used the following estimates. First, by [2, Lemma 3.11], we have

\|A_{\Sigma_{1}}\|^{2}+\sum_{p=2}^{4}\sum_{p=q+1}^{6}\left(A_{\Sigma_{1}}(e_{p},e_{p})A_{\Sigma_{1}}(e_{q},e_{q})-A_{\Sigma_{1}}(e_{p},e_{q})^{2}\right)\geq 0.

Moreover, via direct computations we have

	$\displaystyle\\|A_{\Sigma_{2}}\\|^{2}$	$\displaystyle-\frac{1}{2}H_{\Sigma_{2}}^{2}+\sum_{p=3}^{4}\sum_{q=p+1}^{6}\left(A_{\Sigma_{2}}(e_{p},e_{p})A_{\Sigma_{2}}(e_{q},e_{q})-A_{\Sigma_{2}}(e_{p},e_{q})^{2}\right)$
		$\displaystyle=\frac{1}{2}\sum_{p=3}^{6}(A^{\Sigma_{2}}_{pp})^{2}+\sum_{p=3}^{4}\sum_{q=p+1}^{6}(A_{pq}^{\Sigma_{2}})^{2}+2(A_{56}^{\Sigma_{2}})^{2}-A^{\Sigma_{2}}_{55}A_{66}^{\Sigma_{2}}\geq 0,$

and similarly

	$\displaystyle\\|A_{\Sigma_{3}}\\|^{2}$	$\displaystyle-\frac{1}{2}H_{\Sigma_{3}}^{2}+\sum_{q=5}^{6}\left(A_{\Sigma_{3}}(e_{4},e_{4})A_{\Sigma_{3}}(e_{q},e_{q})-A_{\Sigma_{3}}(e_{4},e_{q})^{2}\right)$
		$\displaystyle=\frac{1}{2}\sum_{p=4}^{6}(A_{pp}^{\Sigma_{3}})^{2}+\sum_{q=5}^{6}(A_{4q}^{\Sigma_{3}})^{2}+2(A_{56}^{\Sigma_{3}})^{2}-A_{55}^{\Sigma_{3}}A_{66}^{\Sigma_{3}}\geq 0.$

By the above computation, we know $\Sigma_{3}$ admits a metric with positive conformal Ricci curvature, and hence it has finite diameter; see Theorem 18. As a direct corollary, the fundamental group of $\Sigma_{3}$ is finite and hence $H_{1}(\Sigma_{3})$ is finite. ∎

We need the following slicing Lemma by Bamler-Li-Mantoulidis [1], which is a generalization of Chodosh-Li [3, Lemma 20].

Lemma 32.

There are $\Sigma_{3,1},\Sigma_{3,2},\dots,\Sigma_{3,k}\subset{\Sigma_{2}}$ pairwise disjoint two-sided stable critical points of $\mathcal{A}_{2}$ so that the manifold with boundary $\hat{\Sigma}_{2}:={\Sigma_{2}}\setminus(\cup_{i=1}^{k}\Sigma_{3,i})$ is connected and has $H_{3}(\partial\hat{{\Sigma}}_{2})\to H_{3}(\hat{{\Sigma}}_{2})$ surjective.

Proof.

Suppose we have constructed $\Sigma_{3,1},\Sigma_{3,2},\cdots,\Sigma_{3,j}$ which are pairwise disjoint two-sided stable critical points of $\mathcal{A}_{2}$ and $M_{j}:=\Sigma_{2}\setminus(\cup_{i=1}^{j}\Sigma_{3,i})$ is connected. If the inclusion map $i:H_{3}(\partial M_{j})\to H_{3}(M_{j})$ is not surjective, then there exists $\Sigma_{3,j+1}$ , a closed connected stable two-sided critical point of $\mathcal{A}_{2}$ , so the induction proceeds. To see that this process eventually terminates, we refer to Bamler-Li-Mantoulidis [1, Lemma 2.5]. ∎

Note that, by Poincaré duality and Lemma 31, we have

H_{2}(\Sigma_{3};\mathbb{Z})=H^{1}(\Sigma_{3};\mathbb{Z})=\text{Hom}(H_{1}(\Sigma_{3};\mathbb{Z});\mathbb{Z})=0.

Given $\Omega\subset\hat{{\Sigma}}_{2}$ , we write $\partial\Omega$ for its topological boundary, and we assume $\partial\Omega$ consists of smooth properly embedded surfaces in $\hat{{\Sigma}}_{2}$ .

Lemma 33.

A connected component of $\hat{{\Sigma}}_{2}\setminus\Omega$ contains exactly one component of $\partial\Omega$ .

Proof.

Assuming the contrary, as in Chodosh-Li [3], there exists $\sigma$ an embedded $S^{1}$ such that $[\sigma]$ is not torsion in $H_{1}(\hat{{\Sigma}}_{2})$ . The long exact sequence in homology for $(\hat{{\Sigma}}_{2},\partial\hat{{\Sigma}}_{2})$ yields:

H_{3}(\partial\hat{{\Sigma}}_{2})\to H_{3}(\hat{{\Sigma}}_{2})\to H_{3}(\hat{{\Sigma}}_{2},\partial\hat{{\Sigma}}_{2})\to H_{2}(\partial\hat{{\Sigma}}_{2})=0.

The final term vanishes since $\partial\hat{{\Sigma}}_{2}$ consisits of components with vanishing second homology group. Combining with Lemma 32, we conclude that $H_{3}(\hat{{\Sigma}}_{2},\partial\hat{{\Sigma}}_{2})=0$ . Poincaré duality implies that $H^{1}(\hat{{\Sigma}}_{2})=0$ , and so the universal coefficient theorem implies that $H_{1}(\hat{{\Sigma}}_{2})$ is torsion. This is a contradiction. ∎

Next, we proceed to the dice-procedure as in Chodosh-Li [3]. We consider the following $\mu$ -bubble functional:

\mathcal{A}_{3}(\Omega)=\int_{\partial^{*}\Omega}uw-\int_{\Omega}(\chi_{\Omega}-\chi_{\Omega_{0}})uwh,

where $h$ satisfies

1+h^{2}(x)-2|\nabla h|\geq 0,

and is to be specified later.

Proposition 34.

Suppose $\Upsilon$ is a component of a stable, free boundary $\mu$ -bubble of $\mathcal{A}_{3}$ . Then there exists a conformal factor such that the corresponding conformal Ricci curvature on $\Upsilon$ is positive and the boundary is minimal after the conformal change.

Proof.

By the first variation, the mean curvature of $\Upsilon$ with respect to the (outer) unit normal $\nu_{\Upsilon}$ of $\Upsilon$ in ${\Sigma_{2}}$ is given by

H_{\Upsilon}=-\langle\nabla_{{\Sigma_{2}}}\log(uw),\nu_{\Upsilon}\rangle+h.

For any smooth function $\psi$ on $\Upsilon$ , the second variation formula gives

	$\displaystyle 0\leq$	$\displaystyle\int_{\Upsilon}\|\nabla_{\Upsilon}\psi\|^{2}uw-\left(\\|A_{\Upsilon}\\|^{2}+\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{\Upsilon},\nu_{\Upsilon})\right)\psi^{2}uw-\langle\nabla_{{\Sigma_{2}}}h,\nu_{\Upsilon}\rangle\psi^{2}uw-h\langle\nabla_{{\Sigma_{2}}}(uw),\nu_{\Upsilon}\rangle\psi^{2}$
		$\displaystyle+\int_{\Upsilon}\left(\Delta_{{\Sigma_{2}}}(uw)-\Delta_{\Upsilon}(uw)\right)\psi^{2}-\int_{\partial\Upsilon}A_{\partial\hat{\Sigma}_{2}}(\omega,\omega)\psi^{2}uw,$

where $\omega$ is the unit normal of $\partial\hat{\Sigma}_{2}$ . Since $\partial\Upsilon$ meets $\partial\hat{\Sigma}_{2}$ orthogonally, $\omega$ is also the unit outer normal of $\partial\Upsilon$ in $\Upsilon$ .

Since the components of $\partial\hat{\Sigma}_{2}$ are $\mu$ -bubbles of $\mathcal{A}_{2}$ , we see from the first variation formula that

H_{\partial\hat{\Sigma}_{2}}=-\langle\nabla_{{\Sigma_{2}}}\log(uw),\omega\rangle,

where $H_{\partial\hat{\Sigma}_{2}}$ is the mean curvature of $\partial\hat{\Sigma}_{2}$ with respect to $\omega$ . Noting $\partial\Upsilon$ meets $\partial\hat{\Sigma}_{2}$ orthogonally, we can denote by $H_{\partial\Upsilon}$ the mean curvature of $\partial\Upsilon$ with respect to $\omega$ , and write

H_{\partial\hat{\Sigma}_{2}}=H_{\partial\Upsilon}+A_{\partial\hat{\Sigma}_{2}}(\omega,\omega).

Since $\omega$ lies in the tangent space of $\Upsilon$ , we rewrite the first variation formula of $\partial\hat{\Sigma}_{2}$ at the intersection points with $\partial\Upsilon$ as

H_{\partial\hat{\Sigma}_{2}}=-\langle\nabla_{\Upsilon}\log(uw),\omega\rangle.

Now we have that

	$\displaystyle\int_{\Upsilon}\|\nabla_{\Upsilon}\psi\|^{2}uw$	$\displaystyle-\int_{\partial\Upsilon}A_{\partial\hat{\Sigma}_{2}}(\omega,\omega)\psi^{2}uw$
	$\displaystyle=$	$\displaystyle\int_{\Upsilon}-uw\psi\Delta_{\Upsilon}\psi-\psi\langle\nabla_{\Upsilon}\psi,\nabla_{\Upsilon}(uw)\rangle$
		$\displaystyle+\int_{\partial\Upsilon}\left(\langle\nabla_{\Upsilon}\psi,\omega\rangle uw+\langle\nabla_{\Upsilon}(uw),\omega\rangle\psi-\langle\nabla_{\Upsilon}(uw),\omega\rangle\psi-A_{\partial\hat{\Sigma}_{2}}(\omega,\omega)\psi uw\right)\psi$
	$\displaystyle=$	$\displaystyle\int_{\Upsilon}-uw\psi\Delta_{\Upsilon}\psi-\psi\langle\nabla_{\Upsilon}\psi,\nabla_{\Upsilon}(uw)\rangle$
		$\displaystyle+\int_{\partial\Upsilon}\left(\langle\nabla_{\Upsilon}(uw\psi),\omega\rangle+\left(H_{\partial\hat{\Sigma}_{2}}-A_{\partial\hat{\Sigma}_{2}}(\omega,\omega)\right)uw\psi\right)\psi$
	$\displaystyle=$	$\displaystyle\int_{\Upsilon}-uw\psi\Delta_{\Upsilon}\psi-\psi\langle\nabla_{\Upsilon}\psi,\nabla_{\Upsilon}(uw)\rangle+\int_{\partial\Upsilon}\left(\langle\nabla_{\Upsilon}(uw\psi),\omega\rangle+H_{\partial\Upsilon}uw\psi\right)\psi.$

Summarizing the above computations, we can rewrite the second variation formula of $\Upsilon$ as

	$\displaystyle 0\leq$	$\displaystyle\int_{\Upsilon}-uw\psi\Delta_{\Upsilon}\psi-\psi\langle\nabla_{\Upsilon}\psi,\nabla_{\Upsilon}(uw)\rangle-\left(\\|A_{\Upsilon}\\|^{2}+\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{\Upsilon},\nu_{\Upsilon})\right)\psi^{2}uw$
		$\displaystyle\int_{\Upsilon}-\langle\nabla_{{\Sigma_{2}}}h,\nu_{\Upsilon}\rangle\psi^{2}uw-h\langle\nabla_{{\Sigma_{2}}}(uw),\nu_{\Upsilon}\rangle\psi^{2}+\left(\Delta_{{\Sigma_{2}}}(uw)-\Delta_{\Upsilon}(uw)\right)\psi^{2}$
		$\displaystyle+\int_{\partial\Upsilon}\left(\langle\nabla_{\Upsilon}(uw\psi),\omega\rangle+H_{\partial\Upsilon}uw\psi\right)\psi.$

The first variation formula implies the following equality

\frac{1}{2}H_{\Upsilon}^{2}\psi^{2}uw=\frac{1}{2}(uw)^{-1}\langle\nabla_{{\Sigma_{2}}}(uw),\nu_{\Upsilon}\rangle^{2}\psi^{2}-h\langle\nabla_{{\Sigma_{2}}}(uw),\nu_{\Upsilon}\rangle\psi^{2}+\frac{1}{2}h^{2}\psi^{2}uw.

As in Chodosh-Li [3], we use the above equality and the fact that

1-2|\nabla_{{\Sigma_{2}}}h|+h^{2}\geq 0

to further simplify the second variation formula as follows:

	$\displaystyle 0\leq$	$\displaystyle\int_{\Upsilon}-uw\psi\Delta_{\Upsilon}\psi-\psi\langle\nabla_{\Upsilon}\psi,\nabla_{\Upsilon}(uw)\rangle-\left(\\|A_{\Upsilon}\\|^{2}+\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{\Upsilon},\nu_{\Upsilon})-\frac{1}{2}H_{\Upsilon}^{2}-\frac{1}{2}\right)\psi^{2}uw$
		$\displaystyle+\int_{\Upsilon}\left(\Delta_{{\Sigma_{2}}}(uw)-\Delta_{\Upsilon}(uw)\right)\psi^{2}-\frac{1}{2}\langle\nabla_{{\Sigma_{2}}}\log(uw),\nu_{\Upsilon}\rangle^{2}\psi^{2}uw+\int_{\partial\Upsilon}\left(\langle\nabla_{\Upsilon}(uw\psi),\omega\rangle+H_{\partial\Upsilon}uw\psi\right)\psi.$

Thus there exists a smooth positive function $v_{2}$ defined on $\Upsilon$ satisfying

(8)		$\displaystyle-v_{2}^{-1}\Delta_{\Upsilon}v_{2}\geq$	$\displaystyle\langle\nabla_{\Upsilon}\log v_{2},\nabla_{\Upsilon}\log(uw)\rangle-(uw)^{-1}\left(\Delta_{{\Sigma_{2}}}(uw)-\Delta_{\Upsilon}(uw)\right)$
(8)			$\displaystyle+\frac{1}{2}\langle\nabla_{{\Sigma_{2}}}\log(uw),\nu_{\Upsilon}\rangle^{2}+\\|A_{\Upsilon}\\|^{2}+\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{\Upsilon},\nu_{\Upsilon})-\frac{1}{2}H_{\Upsilon}^{2}-\frac{1}{2}.$

and it satisfies the following boundary condition on $\partial\Upsilon$

(9)

\displaystyle\langle\nabla_{\Upsilon}(uwv_{2}),\omega\rangle+H_{\partial\Upsilon}uwv_{2}=0.

The boundary condition implies that $\partial\Upsilon$ is minimal under the conformal change $(uwv_{2})^{2}g$ .

By Proposition 28, we have

(10)		$\displaystyle(uw)^{-1}$	$\displaystyle\Delta_{{\Sigma_{2}}}(uw)$
(10)		$\displaystyle\leq$	$\displaystyle\langle\nabla_{{\Sigma_{2}}}\log u,\nabla_{{\Sigma_{2}}}\log w\rangle-\\|A_{\Sigma_{1}}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\\|A_{{\Sigma_{2}}}\\|^{2}-\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}.$

Combining (8) and (10), we obtain

	$\displaystyle(uwv_{2})^{-1}$	$\displaystyle\Delta_{\Upsilon}(uwv_{2})$
	$\displaystyle=$	$\displaystyle(uw)^{-1}\Delta_{\Upsilon}(uw)+v_{2}^{-1}\Delta_{\Upsilon}(v_{2})+2\langle\nabla_{\Upsilon}\log(uw),\nabla_{\Upsilon}\log v_{2}\rangle$
	$\displaystyle\leq$	$\displaystyle(uw)^{-1}\Delta_{{\Sigma_{2}}}(uw)+\langle\nabla_{\Upsilon}\log(uw),\nabla_{\Upsilon}\log v_{2}\rangle-\\|A_{\Upsilon}\\|^{2}-\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{3},\nu_{3})+\frac{1}{2}H_{\Upsilon}^{2}+\frac{1}{2}-\frac{1}{2}\langle\nabla_{{\Sigma_{2}}}\log(uw),\nu_{\Upsilon}\rangle^{2}$
	$\displaystyle\leq$	$\displaystyle\langle\nabla_{{\Sigma_{2}}}\log u,\nu_{\Upsilon}\rangle\langle\nabla_{{\Sigma_{2}}}\log w,\nu_{\Upsilon}\rangle+\langle\nabla_{\Upsilon}\log u,\nabla_{\Upsilon}\log w\rangle-\frac{1}{2}\langle\nabla_{{\Sigma_{2}}}\log(uw),\nu_{\Upsilon}\rangle^{2}+\langle\nabla_{\Upsilon}\log(uw),\nabla_{\Upsilon}\log v_{2}\rangle$
		$\displaystyle-\\|A_{\Sigma_{1}}\\|^{2}-\\|A_{{\Sigma_{2}}}\\|^{2}-\\|A_{\Upsilon}\\|^{2}-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)-\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{\Upsilon},\nu_{\Upsilon})$
		$\displaystyle+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}H_{\Upsilon}^{2}+1,$
	$\displaystyle\leq$	$\displaystyle\langle\nabla_{\Upsilon}\log(uw),\nabla_{\Upsilon}\log v_{2}\rangle+\langle\nabla_{\Upsilon}\log u,\nabla_{\Upsilon}\log w\rangle-\\|A_{\Sigma_{1}}\\|^{2}-\\|A_{{\Sigma_{2}}}\\|^{2}-\\|A_{\Upsilon}\\|^{2}$
		$\displaystyle-\operatorname{Ric}_{\overline{M}}(\eta,\eta)-\operatorname{Ric}_{\Sigma}(\nu,\nu)-\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{\Upsilon},\nu_{\Upsilon})+\frac{1}{2}H_{{\Sigma_{2}}}^{2}+\frac{1}{2}H_{\Upsilon}^{2}+1.$

Supposing $e_{4}$ is a unit vector in $T\Upsilon$ , we have

	$\displaystyle\operatorname{Ric}_{\Upsilon}(e_{4},e_{4})$	$\displaystyle-(uwv_{2})^{-1}\Delta_{\Upsilon}(uwv_{2})+\frac{1}{2}\|\nabla_{\Upsilon}\log(uwv_{2})\|^{2}$
	$\displaystyle\geq$	$\displaystyle\frac{1}{2}\|\nabla_{\Upsilon}\log(uwv_{2})\|^{2}-\langle\nabla_{\Upsilon}\log(uw),\nabla_{\Upsilon}\log v_{2}\rangle-\langle\nabla_{\Upsilon}\log u,\nabla_{\Upsilon}\log w\rangle$
		$\displaystyle+\\|A_{\Sigma_{1}}\\|^{2}+\\|A_{{\Sigma_{2}}}\\|^{2}+\\|A_{\Upsilon}\\|^{2}-\frac{1}{2}H_{{\Sigma_{2}}}^{2}-\frac{1}{2}H_{\Upsilon}^{2}-1$
		$\displaystyle+\operatorname{Ric}_{\overline{M}}(\eta,\eta)+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)+\operatorname{Ric}_{{\Sigma_{2}}}(\nu_{\Upsilon},\nu_{\Upsilon})+\operatorname{Ric}_{\Upsilon}(e_{4},e_{4})$
	$\displaystyle\geq$	$\displaystyle\\|A_{{\Sigma_{1}}}\\|^{2}+\\|A_{\Sigma_{2}}\\|^{2}+\\|A_{\Upsilon}\\|^{2}-\frac{1}{2}H_{{\Sigma_{2}}}^{2}-\frac{1}{2}H_{\Upsilon}^{2}-1$
		$\displaystyle+\operatorname{Ric}_{\Upsilon}(e_{4},e_{4})+\operatorname{Ric}_{{\Sigma_{2}}}(\omega,\omega)+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu)+\operatorname{Ric}_{\overline{M}}(\eta,\eta)$
	$\displaystyle=$	$\displaystyle\\|A_{{\Sigma_{1}}}\\|^{2}+\\|A_{\Sigma_{2}}\\|^{2}+\\|A_{\Upsilon}\\|^{2}-\frac{1}{2}H_{{\Sigma_{2}}}^{2}-\frac{1}{2}H_{\Upsilon}^{2}-1$
		$\displaystyle+C_{4}(e_{1},e_{2},e_{3},e_{4})+\sum_{p=2}^{4}\sum_{q=p+1}^{6}\left(A_{\Sigma_{1}}(e_{p},e_{p})A_{\Sigma_{1}}(e_{q},e_{q})-A_{\Sigma_{1}}(e_{p},e_{q})^{2}\right)$
		$\displaystyle+\sum_{p=3}^{4}\sum_{q=p+1}^{6}\left(A_{{\Sigma_{2}}}(e_{p},e_{p})A_{{\Sigma_{2}}}(e_{q},e_{q})-A_{{\Sigma_{2}}}(e_{p},e_{q})^{2}\right)$
		$\displaystyle+\sum_{q=5}^{6}\left(A_{\Upsilon}(e_{4},e_{4})A_{\Upsilon}(e_{q},e_{q})-A_{\Upsilon}(e_{4},e_{q})^{2}\right)$
	$\displaystyle\geq$	$\displaystyle C_{4}(e_{1},e_{2},e_{3},e_{4})-1\geq\frac{1}{2}.$

We have completed the proof. ∎

Corollary 35.

Suppose $\Upsilon$ is a component of a free boundary $\mu$ -bubble of $\mathcal{A}_{3}$ . Then it has connected boundary and bounded diameter.

Proof.

Given the previous proposition, the diameter estimate follows from Theorem 18, and the fact that the boundary is connected follows from Theorem 22 and (9). ∎

We can summarize the dice procedure as follows.

Proposition 36.

Suppose $p\in\hat{\Sigma}_{2}$ is a fixed interior point, and assume further that $B_{\varepsilon}(p)\subset\subset\hat{\Sigma}_{2}$ . There exists a finite number $k$ and open connected domains $\{\Omega_{i}\}_{i=1}^{k}$ ,

B_{\varepsilon}(p)=\Omega_{1}\subset\Omega_{2},\subset\cdots\subset\Omega_{k}=\hat{\Sigma}_{2},

with the following properties:

(1)

$d_{\hat{\Sigma}_{2}}(\partial\Omega_{i+1},\partial\Omega_{i})\geq\frac{2}{3}\pi$ ;
(2)

Each component of $\Omega_{i+1}\setminus\Omega_{i}$ has diameter at most $10\pi$ ;
(3)

Any component $\Upsilon\subset\partial\Omega_{j}$ has diameter $\operatorname{diam}\Upsilon\leq\pi$ ;
(4)

Each component of $\partial\Omega_{j}$ is either a closed manifold with finite fundamental group or a compact manifold with connected boundary in $\partial\hat{\Sigma}_{2}$ .

Proof.

The proof follows from induction. Suppose we have constructed $\Omega_{1},\Omega_{2},\cdots,\Omega_{j}$ satisfying the above assumptions and $\Omega_{j}\neq\hat{\Sigma}_{2}$ . Smooth $d(\cdot,\Omega_{j})$ to a function $\rho$ such that $d(\cdot,\Omega_{j})\leq\rho\leq\frac{3}{2}d(\cdot,\Omega_{j})$ and $\rho|_{\Omega_{j}}=0$ . Take $h$ to be

h(x)=-\tan\left(\frac{1}{3}(\rho-\pi)-\frac{\pi}{2}\right),

and note that

1+h^{2}-2|\nabla h|\geq 0.

Minimize $\hat{\mathcal{A}}_{3}$ to get a $\mu$ -bubble $\Omega_{j+1}$ .

By definition of $h$ , $\Omega_{j+1}$ satisfies condition $(1)$ . By Lemma 33 and Lemma 31, it satisfies condition $(2)$ . By Corollary 35 and Lemma 31, it also satisfies condition $(3)$ and $(4)$ . Since condition $(1)$ is satisfied by the sets we constructed, the induction process must terminate after finitely many steps. ∎

We now have all the ingredients we need to prove the last case of the Main Theorem 4.

Proof of Theorem 30.

Let $M$ be as in the statement of the theorem. Assume for the sake of contradiction that $M$ admits a metric with positive $4$ -intermediate curvature. We can assume the $4$ -intermediate curvature of $M$ is at least $\frac{3}{2}$ .

Arguing as in the proof of Theorem 27, we obtain $5$ -dimensional $\Sigma_{1}$ and $4$ -dimensional $\Sigma_{2}$ . By Lemma 31, Lemma 32 and Proposition 36, there exists a set of disjoint embedded closed $3$ manifolds $\Sigma_{3,1},\Sigma_{3,2}\cdots,\Sigma_{3,k}\subset\Sigma_{2}$ with $\operatorname{diam}\Sigma_{3,i}\leq\pi$ . Moreover, there exists a set of embedded compact $3$ -manifolds $\Upsilon_{1},\Upsilon_{2},\cdots,\Upsilon_{l}\subset\Sigma_{2}$ such that, $\operatorname{diam}\Upsilon_{j}\leq\pi$ , $\partial\Upsilon_{j}$ is connected and contained in $\cup_{i=1}^{k}\Sigma_{3,i}$ , and the interiors of $\Upsilon_{j}$ are pairwise disjoint with each other and disjoint with each $\Sigma_{3,i}$ . Also, each component $K_{1},K_{2},\cdots,K_{s}$ of

\Sigma_{2}\setminus\left(\left(\cup_{i=1}^{k}\Sigma_{3,i}\right)\cup\left(\cup_{j=1}^{s}\Upsilon_{j}\right)\right)

has diameter bounded by $10\pi$ .

We write the boundary components of $K_{j}$ as $\hat{\Upsilon}_{j}^{1},\hat{\Upsilon}^{2}_{j},\cdots,\hat{\Upsilon}_{j}^{n(j)}$ . By Proposition 14, there exists a positive $R>0$ , independent of $L$ , such that we can fill-in $\hat{\Upsilon}_{j}^{i}$ by $\hat{K}_{j}^{i}$ with extrinsic diameter at most $R$ . Then

K_{j}-\hat{K}_{j}^{1}-\cdots-\hat{K}_{j}^{n(j)}

is a cycle with extrinsic diameter at most $2R+10\pi$ . By Proposition 14, there exists a $5$ -chain $\tilde{\Gamma}_{j}$ with extrinsic diameter bounded by $\tilde{R}$ , which is independent of $L$ , such that

\partial\tilde{\Gamma}_{j}=K_{j}-\hat{K}_{j}^{1}-\cdots-\hat{K}_{j}^{n(j)}.

Note that

\Sigma_{2}-\sum_{j=1}^{s}\partial\tilde{\Gamma}_{j}=\sum_{j=1}^{s}\sum_{i=1}^{n(j)}\hat{K}_{j}^{i}.

Since $\Upsilon_{i}$ has connected boundary, there exists an index $u(i,j)\in\{1,2,\cdots,k\}$ so that $\hat{\Upsilon}_{j}^{i}$ only intersects with $\Sigma_{3,u(i,j)}$ but not any of the other components of $\partial\hat{\Sigma}_{2}$ . We group the $\hat{K}_{j}^{i}$ by $u(i,j)=a$ for $a\in\{1,2,\cdots,k\}$ . Then

\sum_{\{i,j:u(i,j)=a\}}\hat{K}_{j}^{i}

is a cycle of diameter at most $2R+\pi$ . Furthermore,

\partial\left[\sum_{a=1}^{k}\sum_{\{i,j:u(i,j)=a\}}\hat{K}_{j}^{i}\right]=0.

Therefore, by Proposition 14, there exists $\hat{R}>0$ , independent of $L$ , such that there exists a $5$ -chain $\Theta_{a}$ with extrinsic diameter at most $\hat{R}$ satisfying

\partial\Theta_{a}=\sum_{\{i,j:u(i,j)=a\}}\hat{K}_{j}^{i}.

In conclusion, we have

\Sigma_{2}=\partial\left[\sum_{j=1}^{s}\tilde{\Gamma}_{j}+\sum_{a=1}^{k}\Theta_{a}\right],

where each term in the sum has uniform bounded diameter as $L\to\infty$ . A contradiction is achieved. ∎

4.6. The Case $n=6$ and $m=3$

Finally, we prove the case where $n=6$ and $m=3$ . This will complete the proof of Theorem 4. This case is somewhat subtle and requires a delicate analysis.

Theorem 37.

Let $M^{6}$ be a closed manifold. Assume that the universal cover $\overline{M}$ of $M$ satisfies $H_{6}(\overline{M},\mathbb{Z})=H_{5}(\overline{M},\mathbb{Z})=H_{4}(\overline{M},\mathbb{Z})=0$ . Then $M$ does not admit a metric with positive 3-intermediate curvature.

Let $M$ be as in the statement of the theorem. Assume for the sake of contradiction that $M$ admits a metric with positive $3$ -intermediate curvature. By scaling, we can suppose the 3-intermediate curvature of $M$ is at least $3$ . For a fixed large $L>0$ , we apply Proposition 13 to find a geodesic line $\sigma$ in $\overline{M}$ and a closed manifold $\Lambda^{n-2}$ embedded in $\overline{M}$ such that $\Lambda$ is linked with $\sigma$ and $d(\sigma,\Lambda)\geq 2L$ . Let $\Sigma_{1}$ be the area minimizing minimal hypersurface in $\overline{M}$ with $\partial\Sigma_{1}=\Lambda$ . Let $\eta$ be the unit normal to $\Sigma_{1}$ . Since $\Sigma_{1}$ is area minimizing, there exists a function $u>0$ on $\Sigma_{1}$ which satisfies $\Delta_{\Sigma_{1}}u+(|A_{\Sigma_{1}}|^{2}+\operatorname{Ric}_{\overline{M}}(\eta,\eta))u\leq 0$ .

We are now going to construct a $\mu$ -bubble in $\Sigma_{1}$ . The construction depends on a choice of several parameters $a,\tau,\varepsilon,\alpha$ which depend only on $M$ and not on $L$ . First, we select

0.9748\approx\frac{\sqrt{390}}{10}-1<a<1

close enough to 1 so that the following two conditions hold:

(i)

$(1-a^{2})|R_{M}(X,Y,X,Y)|\leq\frac{1}{100}$ for all unit tangent vectors $X,Y$ to $M$ ;

(ii)

The matrix

\begin{pmatrix}\frac{1}{a}-\frac{9}{16}&\frac{1}{2}-\frac{1}{a}&\frac{\sqrt{3}}{8}\\ \\ \frac{1}{2}-\frac{1}{a}&\frac{1}{a}&0\\ \\ \frac{\sqrt{3}}{8}&0&\frac{1}{4}\end{pmatrix}

is positive definite.

To see that (ii) is possible, note that the determinant of this matrix is $-\frac{1}{16}+\frac{1}{16a}$ which is positive for $a<1$ . Also when $a=1$ , by direct computation, the matrix

\begin{pmatrix}\frac{7}{16}&-\frac{1}{2}&\frac{\sqrt{3}}{8}\\ \\ -\frac{1}{2}&1&0\\ \\ \frac{\sqrt{3}}{8}&0&\frac{1}{4}\end{pmatrix}

has eigenvalues

\lambda_{1}=\frac{1}{32}(27+\sqrt{217})>0,\quad\lambda_{2}=\frac{1}{32}(27-\sqrt{217})>0,\quad\lambda_{3}=0.

Therefore, by continuity, a suitable choice of $a$ is possible. Next, we select $a<\tau<1$ and $\alpha>0$ so that the following condition holds:

(iii)

The matrix

\begin{pmatrix}\frac{3}{16}-\frac{3}{4}\tau+\frac{\tau}{a}&\frac{\tau}{2}-\frac{\tau}{a}&\frac{\sqrt{3}}{8}\\ \\ \frac{\tau}{2}-\frac{\tau}{a}&\frac{\tau}{a}-\alpha&0\\ \\ \frac{\sqrt{3}}{8}&0&\tau-\frac{3}{4}\end{pmatrix}

is positive definite.

Since this matrix reduces to the one in condition (ii) when $\tau=1$ and $\alpha=0$ , such a choice is again possible by continuity. Finally, since $\tau<1$ , we can select $\varepsilon>0$ so that

(iv)

$\displaystyle\tau-\left(\frac{3}{4}+\varepsilon\right)\tau^{2}\geq\frac{\tau}{4}.$

The reason for selecting this choice of parameters will become apparent in the proof.

Next, consider the differential equation

\begin{cases}k^{\prime}(t)=-1-\frac{\alpha}{2}k(t)^{2},\\ k(0)=0.\end{cases}

The solution to this ODE is

k(t)=-\frac{2}{\sqrt{\alpha}}\arctan\left(\sqrt{\frac{\alpha}{2}}t\right)

for $t\in(-\beta,\beta)$ where $\beta=\frac{\pi}{2}\sqrt{\frac{2}{\alpha}}$ . Note that $k(t)\to\infty$ as $t\to-\beta$ and that $k(t)\to-\infty$ as $t\to\beta$ .

Define

\rho_{0}(x)=d_{\overline{M}}(x,\sigma),\quad x\in\overline{M}.

Then let $\rho_{1}$ be a smooth approximation to $\rho_{0}$ which satisfies $\|\nabla\rho_{1}(x)\|\leq 2$ for all $x\in\overline{M}$ , $\rho_{1}<1$ on $\sigma$ , and $\rho_{1}>L+\beta+1$ on $\partial\Sigma_{1}=\Lambda$ . Let $\rho_{2}$ be the restriction of $\rho_{1}$ to $\Sigma_{1}$ . Now choose $\delta>0$ sufficiently small so that $L-\beta+\delta$ , $L+2\delta$ , and $L+\beta+\delta$ are all regular values of $\rho_{2}$ . We can further ensure that $\Sigma_{1}\cap B_{\overline{M}}(\sigma,L/4)\subset\{\rho_{2}\leq L-\beta-1\}$ . Now define

\rho(x)=\rho_{2}(x)-L-\delta,\quad x\in\Sigma_{1}.

Then we have $\|\nabla_{\Sigma_{1}}\rho(x)\|\leq 2$ for all $x\in\Sigma_{1}$ . We define

	$\displaystyle\tilde{\Sigma}_{1}=\{-\beta\leq\rho\leq\beta\},$
	$\displaystyle\partial_{+}\tilde{\Sigma}_{1}=\{\rho=-\beta\},$
	$\displaystyle\partial_{-}\tilde{\Sigma}_{1}=\{\rho=\beta\},$
	$\displaystyle\Omega_{0}=\{-\beta\leq\rho<\delta\}.$

h(x)=k(\rho(x)).

Then $h$ is a smooth function on the interior of $\tilde{\Sigma}_{1}$ which satisfies $h(x)\to\pm\infty$ as $x\to\partial_{\pm}\tilde{\Sigma}_{1}$ . Moreover, we have

\|\nabla_{\Sigma_{1}}h(x)\|=|k^{\prime}(\rho(x))|\cdot\|\nabla_{\Sigma_{1}}\rho(x)\|\leq 2\left(1+\frac{\alpha}{2}k(\rho(x))^{2}\right)=2+\alpha h^{2},

for all $x\in\tilde{\Sigma}_{1}$ . It follows that

(11)

2+\alpha h^{2}-\|\nabla_{\Sigma_{1}}h\|\geq 0

on $\tilde{\Sigma}_{1}$ . Proposition 15 (with $u$ replaced by $u^{a}$ ) implies that there exists a minimizer $\Omega\subset\tilde{\Sigma}_{1}$ of the $\mu$ -bubble functional

\mathcal{A}(\Omega)=\int_{\partial\Omega}u^{a}\,d\mathcal{H}^{4}-\int_{\tilde{\Sigma}_{1}}(\chi_{\Omega}-\chi_{\Omega_{0}})u^{a}h\,d\mathcal{H}^{5}

such that $\Omega\operatorname{\Delta}\Omega_{0}$ is compactly contained in the interior of $\tilde{\Sigma}_{1}$ . Let $\Sigma_{2}$ be a component of $\partial\Omega$ .

According to Proposition 16 with $u$ replaced by $u^{a}$ (see also Mazet [14] Section 4.2), the first variation satisfies

H_{\Sigma_{2}}=h-au^{-1}\langle\nabla_{\Sigma_{1}}u,\nu\rangle,

and the second variation formula implies

	$\displaystyle 0$	$\displaystyle\leq\int_{\Sigma_{2}}u^{a}\bigg{[}\|\nabla_{\Sigma_{2}}\psi\|^{2}-(\|A_{\Sigma_{2}}\|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu))\psi^{2}-au^{-2}\langle\nabla_{\Sigma_{1}}u,\nu\rangle^{2}\psi^{2}$
		$\displaystyle\qquad\qquad+au^{-1}(\Delta_{\Sigma_{1}}u-\Delta_{\Sigma_{2}}u-H_{\Sigma_{2}}\langle\nabla_{\Sigma_{1}}u,\nu\rangle)\psi^{2}-\langle\nabla_{\Sigma_{1}}h,\nu\rangle\psi^{2}\bigg{]}$

for all $\psi\in C^{1}(\Sigma_{2})$ . It follows that there exists a function $w>0$ on $\Sigma_{2}$ which satisfies

	$\displaystyle\operatorname{div}_{\Sigma_{2}}(u^{a}\nabla_{\Sigma_{2}}w)$	$\displaystyle\leq\bigg{[}-(\|A_{\Sigma_{2}}\|^{2}+\operatorname{Ric}_{\Sigma_{1}}(\nu,\nu))-a(\|A_{\Sigma_{1}}\|^{2}+\operatorname{Ric}_{\overline{M}}(\eta,\eta))\bigg{]}u^{a}w$
		$\displaystyle\qquad+\bigg{[}-\langle\nabla_{\Sigma_{1}}h,\nu\rangle-au^{-2}\langle\nabla_{\Sigma_{1}}u,\nu\rangle^{2}-aH_{\Sigma_{2}}u^{-1}\langle\nabla_{\Sigma_{1}}u,\nu\rangle-a\frac{\Delta_{\Sigma_{2}}u}{u}\bigg{]}u^{a}w.$

Now we turn our attention to the diameter bound.

Proposition 38.

The diameter of $\Sigma_{2}$ is uniformly bounded independently of $L$ .

Proof.

We are going to apply the Shen-Ye generalized Bonnet-Myers theorem with test function $u^{a}w$ . Fix an orthonormal basis $\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}$ for $\overline{M}$ so that $e_{1}=\eta$ and $e_{2}=\nu$ . By condition (iv), it suffices to show that

\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\tau\frac{\Delta_{\Sigma_{2}}(u^{a}w)}{u^{a}w}+\frac{\tau}{4}|\nabla_{\Sigma_{2}}\ln(u^{a}w)|^{2}\geq\kappa>0

for some $\kappa$ which does not depend on $L$ . We have

	$\displaystyle\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\tau\frac{\Delta_{\Sigma_{2}}(u^{a}w)}{u^{a}w}+\frac{\tau}{4}\|\nabla_{\Sigma_{2}}\ln(u^{a}w)\|^{2}$
	$\displaystyle\qquad=\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\tau\frac{\operatorname{div}_{\Sigma_{2}}(u^{a}\nabla_{\Sigma_{2}}w)}{u^{a}w}-\tau\frac{\operatorname{div}_{\Sigma_{2}}(w\nabla_{\Sigma_{2}}u^{a})}{u^{a}w}+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}.$

Now observe that

	$\displaystyle-\tau\frac{\operatorname{div}_{\Sigma_{2}}(w\nabla_{\Sigma_{2}}u^{a})}{u^{a}w}+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}$
	$\displaystyle\qquad\quad=-\tau\left[\left\langle\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}},\frac{\nabla_{\Sigma_{2}}w}{w}\right\rangle+a(a-1)\frac{\|\nabla_{\Sigma_{2}}u\|^{2}}{u^{2}}+a\frac{\Delta_{\Sigma_{2}}u}{u}\right]+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}$
	$\displaystyle\qquad\quad\geq-a\tau\frac{\Delta_{\Sigma_{2}}u}{u}-\tau\left\langle\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}},\frac{\nabla_{\Sigma_{2}}w}{w}\right\rangle+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}$
	$\displaystyle\qquad\quad=-a\tau\frac{\Delta_{\Sigma_{2}}u}{u}+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}-\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}\geq-a\tau\frac{\Delta_{\Sigma_{2}}u}{u},$

where we used the fact that $a<1$ . Thus we have

	$\displaystyle\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})-\tau\frac{\Delta_{\Sigma_{2}}(u^{a}w)}{u^{a}w}+\frac{\tau}{4}\|\nabla_{\Sigma_{2}}\ln(u^{a}w)\|^{2}$
	$\displaystyle\qquad\quad\geq\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})+\tau\|A_{\Sigma_{2}}\|^{2}+\tau\operatorname{Ric}_{\Sigma_{1}}(e_{2},e_{2})+a\tau\|A_{\Sigma_{1}}\|^{2}+a\tau\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})$
	$\displaystyle\qquad\qquad\qquad+\tau\langle\nabla_{\Sigma_{1}}h,e_{2}\rangle+a\tau H_{\Sigma_{2}}u^{-1}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle+a\tau u^{-2}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle^{2}$
	$\displaystyle\qquad\quad=\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})+\operatorname{Ric}_{\Sigma_{1}}(e_{2},e_{2})+\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})+\mathcal{R}$
	$\displaystyle\qquad\qquad\qquad+\tau\|A_{\Sigma_{2}}\|^{2}+a\tau\|A_{\Sigma_{1}}\|^{2}+(\tau-1)\sum_{q=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{qq}-(A^{\Sigma_{1}}_{2q})^{2}\right]$
	$\displaystyle\qquad\qquad\qquad+\tau\langle\nabla_{\Sigma_{1}}h,e_{2}\rangle+a\tau H_{\Sigma_{2}}u^{-1}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle+a\tau u^{-2}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle^{2},$

where $\mathcal{R}$ denotes a sum of 9 terms of the form $(\tau-1)R_{\overline{M}}(X,Y,X,Y)$ or $(a\tau-1)R_{\overline{M}}(X,Y,X,Y)$ and therefore satisfies $|\mathcal{R}|\leq\frac{1}{2}$ by condition (i) on $a$ .

Define the quantity

	$\displaystyle\mathcal{K}:=$	$\displaystyle\operatorname{Ric}_{\Sigma_{2}}(e_{3},e_{3})+\operatorname{Ric}_{\Sigma_{1}}(e_{2},e_{2})+\operatorname{Ric}_{M}(e_{1},e_{1})$
		$\displaystyle\quad+\tau\|A_{\Sigma_{2}}\|^{2}+a\tau\|A_{\Sigma_{1}}\|^{2}+(\tau-1)\sum_{q=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{qq}-(A^{\Sigma_{1}}_{2q})^{2}\right]$
		$\displaystyle\quad+\tau\langle\nabla_{\Sigma_{1}}h,e_{2}\rangle+a\tau H_{\Sigma_{2}}u^{-1}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle+a\tau u^{-2}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle^{2}.$

By the bound on $\mathcal{R}$ , to prove the proposition, it suffices to show that $\mathcal{K}\geq 1$ . According to [2, Lemma 3.8], we have

(12)

\displaystyle\mathcal{K}=C_{3}(e_{1},e_{2},e_{3})+\mathcal{B}_{1}+\mathcal{B}_{2}+\tau\langle\nabla_{\Sigma_{1}}h,e_{2}\rangle,

where

\mathcal{B}_{1}=a\tau|A_{\Sigma_{1}}|^{2}+\tau\sum_{q=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{qq}-(A^{\Sigma_{1}}_{2q})^{2}\right]+\sum_{q=4}^{6}\left[A^{\Sigma_{1}}_{33}A^{\Sigma_{1}}_{qq}-(A^{\Sigma_{1}}_{3q})^{2}\right]

and

\mathcal{B}_{2}=\tau|A_{\Sigma_{2}}|^{2}+\sum_{q=4}^{6}\left[A^{\Sigma_{2}}_{33}A^{\Sigma_{2}}_{qq}-(A^{\Sigma_{2}}_{pq})^{2}\right]+a\tau H_{\Sigma_{2}}u^{-1}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle+a\tau u^{-2}\langle\nabla_{\Sigma_{1}}u,e_{2}\rangle^{2}.

Next we focus on obtaining lower bounds for $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ .

Lemma 39.

The quantity $\mathcal{B}_{1}$ is non-negative.

Proof.

Dropping the subscript and superscript $\Sigma_{1}$ ’s, we have

\displaystyle\mathcal{B}_{1}=a\tau|A|^{2}+\tau\sum_{q=3}^{6}\left[A_{22}A_{qq}-A_{2q}^{2}\right]+\sum_{q=4}^{6}\left[A_{33}A_{qq}-A_{3q}^{2}\right].

Since $\Sigma_{1}$ is minimal and $a\tau>a^{2}>1/2$ , it follows that

	$\displaystyle\mathcal{B}_{1}$	$\displaystyle\geq a\tau(A_{22}^{2}+A_{33}^{2}+A_{44}^{2}+A_{55}^{2}+A_{66}^{2})+\tau(A_{22}A_{33}+A_{22}A_{44}+A_{22}A_{55}+A_{22}A_{66})$
		$\displaystyle\qquad+(A_{33}A_{44}+A_{33}A_{55}+A_{33}A_{66})$
		$\displaystyle=(a\tau-\frac{1}{2})(A_{22}^{2}+A_{33}^{2}+A_{44}^{2}+A_{55}^{2}+A_{66}^{2})+\frac{1}{2}(A_{22}+A_{33}+A_{44}+A_{55}+A_{66})^{2}$
		$\displaystyle\qquad+(\tau-1)(A_{22}A_{33}+A_{22}A_{44}+A_{22}A_{55}+A_{22}A_{66})-A_{44}A_{55}-A_{44}A_{66}-A_{55}A_{66}$
		$\displaystyle=(a\tau-\frac{1}{2})(A_{22}^{2}+A_{33}^{2}+A_{44}^{2}+A_{55}^{2}+A_{66}^{2})$
		$\displaystyle\qquad+(\tau-1)(A_{22}A_{33}+A_{22}A_{44}+A_{22}A_{55}+A_{22}A_{66})-A_{44}A_{55}-A_{44}A_{66}-A_{55}A_{66}.$

Again, since $\Sigma_{1}$ is minimal, we have

A_{22}^{2}+A_{33}^{2}\geq\frac{1}{2}(A_{22}+A_{33})^{2}=\frac{1}{2}(A_{44}+A_{55}+A_{66})^{2}.

Thus we obtain

	$\displaystyle\mathcal{B}_{1}$	$\displaystyle\geq 2(1-\tau)(A_{22}^{2}+A_{33}^{2}+A_{44}^{2}+A_{55}^{2}+A_{66}^{2})+(\tau-1)(A_{22}A_{33}+A_{22}A_{44}+A_{22}A_{55}+A_{22}A_{66})$
		$\displaystyle\qquad+\frac{3}{2}(2\tau-\frac{5}{2}+a\tau)(A_{44}^{2}+A_{55}^{2}+A_{66}^{2})+(2\tau-\frac{7}{2}+a\tau)(A_{44}A_{55}+A_{44}A_{66}+A_{55}A_{66})$
		$\displaystyle\geq\frac{3}{2}(2\tau-\frac{5}{2}+a\tau)(A_{44}^{2}+A_{55}^{2}+A_{66}^{2})+(2\tau-\frac{7}{2}+a\tau)(A_{44}A_{55}+A_{44}A_{66}+A_{55}A_{66}).$

This will be non-negative as long as

\frac{3}{2}(2\tau-\frac{5}{2}+a\tau)\geq\frac{7}{2}-2\tau-a\tau.

This is equivalent to

5\tau+\frac{5}{2}\tau a\geq\frac{29}{4},

which holds since $\tau>a$ and $a>\frac{\sqrt{390}}{10}-1$ . ∎

Lemma 40.

The quantity $\mathcal{B}_{2}$ satisfies $\mathcal{B}_{2}\geq\alpha h^{2}$ .

Proof.

Using $H_{\Sigma_{2}}=h-au^{-1}\langle\nabla_{\Sigma_{1}}u,\eta\rangle$ , and dropping the subscript and superscript $\Sigma_{2}$ ’s, we have

\displaystyle\mathcal{B}_{2}-\alpha h^{2}=\tau|A|^{2}+\sum_{q=4}^{6}\left[A_{33}A_{qq}-(A_{pq})^{2}\right]+\tau H(h-H)+\frac{\tau}{a}(H-h)^{2}-\alpha h^{2}.

Now we analyze this quadratic form as in Mazet [14]. Write

A=\begin{pmatrix}\frac{H}{4}+\Phi_{33}&A_{34}&A_{35}&A_{36}\\ A_{43}&\frac{H}{4}+\Phi_{44}&A_{45}&A_{46}\\ A_{53}&A_{54}&\frac{H}{4}+\Phi_{55}&A_{56}\\ A_{63}&A_{64}&A_{65}&\frac{H}{4}+\Phi_{66}\end{pmatrix}.

Then, since $\tau\geq\frac{1}{2}$ , we have

	$\displaystyle\tau\|A\|^{2}+\sum_{q=4}^{6}\left[A_{33}A_{qq}-(A_{pq})^{2}\right]+\tau H(h-H)+\frac{\tau}{a}(H-h)^{2}-\alpha h^{2}$
	$\displaystyle\qquad\geq\tau\frac{H^{2}}{4}+\tau\|\Phi\|^{2}+A_{33}(H-A_{33})+\tau H(h-H)+\frac{\tau}{a}(H-h)^{2}-\alpha h^{2}$
	$\displaystyle\qquad=\tau\frac{H^{2}}{4}+\tau\|\Phi\|^{2}+(\frac{H}{4}+\Phi_{33})(\frac{3}{4}H-\Phi_{33})+\tau H(h-H)+\frac{\tau}{a}(H-h)^{2}-\alpha h^{2}$
	$\displaystyle\qquad=\left(\frac{3}{16}-\frac{3}{4}\tau+\frac{\tau}{a}\right)H^{2}+\left(\frac{\tau}{a}-\alpha\right)h^{2}+\tau\|\Phi\|^{2}-\Phi_{33}^{2}+\left(\tau-\frac{2\tau}{a}\right)Hh+\frac{1}{2}H\Phi_{33}.$

Let $E=\{(x_{3},x_{4},x_{5},x_{6})\in\mathbb{R}^{4}:x_{3}+x_{4}+x_{5}+x_{6}=0\}$ . Then any vector in $E$ can be written in the form

\frac{1}{\sqrt{2}}\begin{pmatrix}0\\ 0\\ 1\\ -1\end{pmatrix}y_{1}+\frac{1}{\sqrt{6}}\begin{pmatrix}0\\ 2\\ -1\\ -1\end{pmatrix}y_{2}+\frac{1}{\sqrt{12}}\begin{pmatrix}3\\ -1\\ -1\\ -1\end{pmatrix}z.

Expressing $\Phi$ in these coordinates, the previous quadratic form is at least

\left(\frac{3}{16}-\frac{3}{4}\tau+\frac{\tau}{a}\right)H^{2}+\left(\frac{\tau}{a}-\alpha\right)h^{2}+\left(\tau-\frac{3}{4}\right)z^{2}+\left(\tau-\frac{2\tau}{a}\right)Hh+\frac{\sqrt{3}}{4}Hz.

Condition (iii) on $a$ , $\tau$ , and $\alpha$ ensures that this quadratic form in $H$ , $h$ , and $z$ is positive definite. The lemma follows. ∎

Combining (12) with Lemma 39, Lemma 40, the assumption on the intermediate curvature, and (11) we now obtain

\mathcal{K}\geq 3+\alpha h^{2}-\tau\|\nabla_{\Sigma_{1}}h\|\geq 1.

Proposition 38 follows. ∎

Finally, we can finish the proof of Theorem 37. Since $\partial{\Sigma_{2}}=0$ and $\operatorname{diam}({\Sigma_{2}})$ is uniformly bounded and $H_{4}(\overline{M},\mathbb{Z})=0$ , it follows from Proposition 14 that ${\Sigma_{2}}$ bounds within an $R$ -neighborhood for some constant $R$ that depends only on $\overline{M}$ . Applying this argument to each component of $\partial\Omega$ , we see that $\partial\Omega$ bounds within its $R$ -neighborhood in $\overline{M}$ . Since $d(\partial\Omega,\sigma)\geq L/4$ and $\partial\Omega$ is linked with $\sigma$ by construction, this is a contradiction provided we select $L>2R$ . This completes the proof.

5. Mapping Version and Classification

In this section we prove a mapping version and a refinement of Theorem 4 into a positive result. We start with the lifting Lemma proven by Chodosh-Li-Liokumovich [4].

Lemma 41 (Chodosh-Li-Liokumovich).

Suppose that $X,N$ are closed oriented manifolds and $f:N\to X$ has non-zero degree. Letting $\bar{X}$ denote the universal covering of $X$ , there exists a connected cover $\hat{N}\to N$ and a lift $\hat{f}:\hat{N}\to\bar{X}$ such that $\hat{f}$ is proper and $\deg\hat{f}=\deg f$ .

Remark 42.

It can be checked that $\hat{f}$ is a Lipschitz map.

Corollary 43 (Mapping Version).

Suppose $M^{n}$ is a closed manifold and that the universal cover $\overline{M}$ of $M$ satisfies

H_{n}(\overline{M},\mathbb{Z})=H_{n-1}(\overline{M},\mathbb{Z})=\ldots=H_{n-m+1}(\overline{M},\mathbb{Z})=0.

Assume further $N^{n}$ is a closed manifold with non-zero degree map to $M^{n}$ , then $N$ does not admit a metric of positive $m$ -intermediate curvature in any of the following cases: $n\in\{3,4,5\}$ and $m\in\{1,2,\dots,n-1\}$ ; $n=6$ and $m\in\{1,2,{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}3},4\}$ ; $n$ is arbitrary and $m=1$ .

Proof.

Since $\hat{f}$ is proper with non-zero degree and since it is a Lipschitz map, all the diameter estimate arguments are true in the connected cover $\hat{N}$ given by Lemma 41, the result then follows. ∎

We now summarize the filling estimates for $n=6,m=4$ in previous sections as follows.

Theorem 44 (Filling Estimates).

Suppose $(N^{6},g)$ is a closed manifold with positive $4$ -intermediate curvature no less than $\frac{3}{2}$ . Fix a connected Riemannian cover $\hat{N}$ :

Consider a closed embedded $4$ -manifold $\hat{\Lambda}$ such that $[\hat{\Lambda}]=0\in H_{4}(\hat{N};\mathbb{Z})$ . Then there exists a $5$ -chain $\hat{\Sigma}\subset B_{L}(\hat{\Lambda})$ and a $4$ -dimensional submanifold $\hat{\Gamma}$ such that

\partial\hat{\Sigma}=\hat{\Lambda}-\hat{\Gamma}

as chains.

Furthermore, there are $4$ -chains $\hat{K}_{1},\cdots,\hat{K}_{s}$ with diameter bounded by $L$ and $3$ -cycles $\{\hat{\Upsilon}_{j}^{l}\}$ where $j=1,2,\cdots,s$ and $l=1,2,\cdots,n(j)$ such that

\hat{\Gamma}=\sum_{j=1}^{s}\hat{K}_{j},\quad\partial\hat{K}_{j}=\sum_{l=1}^{n(j)}\hat{\Upsilon}_{j}^{l},

where $\Upsilon_{j}^{l}$ has diameter bounded by $L$ , and the above equalities hold as chains.

Finally, there is an integer $q$ and a function

u:\{(j,l):j=1,2,\cdots,s,l=1,\cdots,n(j)\}\to\{1,2,\cdots,q\}

such that for any $t\in\{1,2,\cdots,q\}$ ,

\operatorname{diam}\left(\cup_{(j,l)\in u^{-1}(t)}\hat{\Upsilon}_{j}^{l}\right)\leq L

and

\sum_{u(j,l)=u^{-1}(r)}\hat{\Upsilon}_{j}^{l}=0

as chains.

Theorem 45.

Suppose $N^{n}$ is a closed manifold which admits positive $m$ -intermediate curvature. Further suppose that either

(1)

$n=5$ , $m=2$ , and $\pi_{2}(N^{5})=0$ ; or
(2)

$n=6$ , $m=4$ , and $\pi_{2}(N^{6})=\pi_{3}(N^{6})=\pi_{4}(N^{6})=0$ .

Then a finite covering of $N$ is homeomorphic to $S^{n}$ or connected sum of $S^{n-1}\times S^{1}$ .

Proof.

The idea is to prove the fundamental group of $N$ is virtually free, and then with the aid of Theorem 1.3 in Gadgil-Seshadri [5], we obtain the desired result.

Case 1: $n=5,m=2$ .

By Ma [13, Theorem 1.6], $\pi_{1}(N^{5})$ is virtually free. Thus there exists a finite connected cover $\hat{N}$ of $N$ such that $\pi_{1}(\hat{N})$ is free. Since $\pi_{2}(N)=0$ , by Gadgil- Seshadri [5, Theorem 1.3], the proof then follows.

Case 2: $n=6,m=4$ . Since $\pi_{2}(N)=\pi_{3}(N)=\pi_{4}(N)=0$ , by the Hurewicz Theorem, the universal covering $\tilde{N}$ of $N$ has trivial $4$ th homology group. Then as a direct Corollary of Theorem 44 and Proposition 14, there exists an $L=L(N,g)>0$ such that for a closed embedded $4$ -submanifold in $\tilde{N}$ is null-homologous in its $L$ -neighborhood. By Chodosh-Li-Liokumovich [4, Proposition 8], for any point $p\in\tilde{N}$ , each connected component of a level set of $d(p,\cdot)$ has diameter bounded by $20L$ . By Chosdosh-Li-Liokumovich [4, Corollary 14], $\pi_{1}(N)$ is virtually free. Since $\pi_{2}(N)=\pi_{3}(N)=0$ , by Gadgil- Seshadri [5, Theorem 1.3], the proof then follows. ∎

With Lemma 41, we have the following mapping version classification.

Corollary 46.

Suppose $N^{n}$ is a closed manifold admits positive $m$ -intermediate curvature, assume further there exists a non-zero degree map $f:N\to X$ , where $X$ is a closed manifold satisfying

(1)

when $n=5$ , $m=2$ , $\pi_{2}(X^{5})=0$ ;
(2)

when $n=6$ , $m=4$ , $\pi_{2}(X^{6})=\pi_{3}(X^{6})=\pi_{4}(X^{6})=0$ .

Then a finite covering of $N$ is homeomorphic to $S^{n}$ or connected sum of $S^{n-1}\times S^{1}$ .

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	$\displaystyle\operatorname{Ric}_{\Sigma_{1}}$	$\displaystyle(e_{2},e_{2})-\tau f^{-1}\Delta_{\Sigma_{1}}f\phantom{\int}$
		$\displaystyle\geq\operatorname{Ric}_{\Sigma_{1}}(e_{2},e_{2})+\tau\|A_{\Sigma_{1}}\|^{2}+\tau\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})$
		$\displaystyle=\operatorname{Ric}_{\overline{M}}(e_{2},e_{2})+\tau\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})-R_{\overline{M}}(e_{1},e_{2},e_{1},e_{2})+\tau\|A_{\Sigma_{1}}\|^{2}+\sum_{j=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{jj}-(A^{\Sigma_{1}}_{2j})^{2}\right]$
		$\displaystyle\geq 2+(\tau-1)\operatorname{Ric}_{\overline{M}}(e_{1},e_{1})+\tau\|A_{\Sigma_{1}}\|^{2}+\sum_{j=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{jj}-(A^{\Sigma_{1}}_{2j})^{2}\right]$
		$\displaystyle\geq 1+\tau\|A_{\Sigma_{1}}\|^{2}+\sum_{j=3}^{6}\left[A^{\Sigma_{1}}_{22}A^{\Sigma_{1}}_{jj}-(A^{\Sigma_{1}}_{2j})^{2}\right].$

	$\displaystyle-\tau\frac{\operatorname{div}_{\Sigma_{2}}(w\nabla_{\Sigma_{2}}u^{a})}{u^{a}w}+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}$
	$\displaystyle\qquad\quad=-\tau\left[\left\langle\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}},\frac{\nabla_{\Sigma_{2}}w}{w}\right\rangle+a(a-1)\frac{\|\nabla_{\Sigma_{2}}u\|^{2}}{u^{2}}+a\frac{\Delta_{\Sigma_{2}}u}{u}\right]+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}$
	$\displaystyle\qquad\quad\geq-a\tau\frac{\Delta_{\Sigma_{2}}u}{u}-\tau\left\langle\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}},\frac{\nabla_{\Sigma_{2}}w}{w}\right\rangle+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}+\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}$
	$\displaystyle\qquad\quad=-a\tau\frac{\Delta_{\Sigma_{2}}u}{u}+\frac{\tau}{4}\left\|\frac{\nabla_{\Sigma_{2}}u^{a}}{u^{a}}-\frac{\nabla_{\Sigma_{2}}w}{w}\right\|^{2}\geq-a\tau\frac{\Delta_{\Sigma_{2}}u}{u},$

	$\displaystyle\tau\|A\|^{2}+\sum_{q=4}^{6}\left[A_{33}A_{qq}-(A_{pq})^{2}\right]+\tau H(h-H)+\frac{\tau}{a}(H-h)^{2}-\alpha h^{2}$
	$\displaystyle\qquad\geq\tau\frac{H^{2}}{4}+\tau\|\Phi\|^{2}+A_{33}(H-A_{33})+\tau H(h-H)+\frac{\tau}{a}(H-h)^{2}-\alpha h^{2}$
	$\displaystyle\qquad=\tau\frac{H^{2}}{4}+\tau\|\Phi\|^{2}+(\frac{H}{4}+\Phi_{33})(\frac{3}{4}H-\Phi_{33})+\tau H(h-H)+\frac{\tau}{a}(H-h)^{2}-\alpha h^{2}$
	$\displaystyle\qquad=\left(\frac{3}{16}-\frac{3}{4}\tau+\frac{\tau}{a}\right)H^{2}+\left(\frac{\tau}{a}-\alpha\right)h^{2}+\tau\|\Phi\|^{2}-\Phi_{33}^{2}+\left(\tau-\frac{2\tau}{a}\right)Hh+\frac{1}{2}H\Phi_{33}.$

On the topology of manifolds with positive intermediate curvature

Abstract.

1. Introduction

Theorem 1.

Theorem 2 (Brendle-Hirsch-Johne [2]).

Conjecture 3.

Theorem 4 (Main Theorem).

Corollary 5.

Theorem 6.

Theorem 7.

1.1. Further Discussion and Proof Ideas

Definition 8.

Theorem 9.

Remark 10.

1.2. Organization

1.3. Acknowledgments

2. Preliminaries

Definition 11.

2.1. Topological Preliminaries

Lemma 12.

Proof.

Proposition 13.

Proof.

Proposition 14.

Proof.

2.2. Preliminaries on μ\mu-bubbles

Proposition 15 (Gromov[9], Zhu [20]).

Proposition 16 (Chodosh-Li [3]).

2.3. Diameter Estimates

Theorem 17 (Shen-Ye [17]).

Theorem 18 (Shen-Ye [18]).

Definition 19.

Theorem 20 (Shen-Ye [18]).

3. Frankel Type Theorems

Theorem 21.

Proof.

Theorem 22.

Proof.

4. Proof of the Main Theorem

4.1. The Ricci Curvature Case

Theorem 23.

Proof.

4.2. The Scalar Curvature Case

Theorem 24.

Proof.

Corollary 25.

4.3. The Codimension Two Case

Theorem 26.

Proof.

4.4. The Case n=5n=5 and m=3m=3

Theorem 27.

Proposition 28.

Proof.

Proposition 29.

Proof.

4.5. The Case n=6n=6 and m=4m=4

Theorem 30.

Lemma 31.

Proof.

Lemma 32.

Proof.

Lemma 33.

Proof.

Proposition 34.

Proof.

Corollary 35.

Proof.

Proposition 36.

Proof.

Proof of Theorem 30.

4.6. The Case n=6n=6 and m=3m=3

Theorem 37.

Proposition 38.

Proof.

Lemma 39.

Proof.

Lemma 40.

Proof.

5. Mapping Version and Classification

Lemma 41 (Chodosh-Li-Liokumovich).

2.2. Preliminaries on $\mu$ -bubbles

4.4. The Case $n=5$ and $m=3$

4.5. The Case $n=6$ and $m=4$

4.6. The Case $n=6$ and $m=3$