scalar curvature rigidity of the four-dimensional sphere

Extremality and rigidity properties of Riemannian manifolds with lower scalar curvature bounds have been the subject of intensive study in recent years. For a comprehensive overview of the subject, we refer to Gromov’s Four lectures on scalar curvature [gromov-four-lectures]. A cornerstone result in comparison geometry with scalar curvature is the rigidity of the round sphere in the spin setting, established by Llarull. Throughout this paper, we denote by $g_{\mathbb{S}^{n}}$ the standard round metric on the $n$-dimensional sphere $\mathbb{S}^{n}$.

This result illustrates the beautiful interplay between metric, curvature and topological information in scalar curvature geometry. Its proof relies on the Dirac operator method, requiring the hypothesis that $M$ is spin. A big open question in the field is whether the spin assumption can be dispensed with in Theorem 1.1. In this paper, we address this question affirmatively, at least in dimension four.

Our approach intertwines various techniques from geometric analysis: minimal hypersurfaces, Ricci flow, and harmonic map heat flow. A key tool in our method is the utilization of $\mu$-bubbles, that are stable solutions to prescribed mean curvature problems. This technique, pioneered by Gromov [Gromov-macroscopic-dimension, Section 5$\nicefrac{{5}}{{6}}$], has been successfully used in addressing some challenging questions in scalar curvature geometry. Examples of its applications can be found in [li-polyhedron, chodosh2023generalized, gromov2020metrics, zhu_width_mu, chodosh-sufficiently-connected, cecchini-two-ends]. Drawing from Gromov’s ideas [gromov-four-lectures, Section 5], we utilize this technique to rule out the case where all the inequalities in A are strict. However, directly proving extremality and rigidity using $\mu$-bubbles poses a significant challenge. To overcome this difficulty, we exploit the monotonicity of the harmonic map heat flow coupled with the Ricci flow, recently established by Lee and Tam [LeeTam]. This approach enables us to reduce A to the situation where all the inequalities are strict, except when the metric is Einstein. Remarkably, Llarull’s rigidity theorem for Einstein manifolds follows from classical comparison geometry.

To further illustrate our strategy, let us compare it with Gromov’s perspective [gromov-four-lectures, Section 5.7]. We regard $\mathbb{S}^{n}$ with two antipodal points removed as a warped product over $\mathbb{S}^{n-1}$. From this viewpoint, Theorem 1.1 becomes a question about the scalar curvature rigidity of the degenerate spherical band

In the spin setting, this question has been recently addressed independently by Bär-Brendle-Hanke-Wang [Baer:2023aa] and by the second and third authors [wang2023scalar] utilizing the Dirac operator method. Gromov outlined an alternative approach to this problem employing $\mu$-bubbles. The crucial observation [gromov-four-lectures, Section 5.5] is that the variational formulas for $\mu$-bubbles, in this context, are related to the following stronger version of Theorem 1.1, due to Listing.

Employing the outlined strategy to study the degenerate spherical band (1.1) poses two main challenges. Firstly, constructing suitable $\mu$-bubbles on open incomplete manifolds presents significant difficulties. While in dimension three this construction has been successfully carried out by Hu, Liu, and Shi [hu-3d-spherical] leveraging the extra control provided by the Gauss-Bonnet theorem, how to extend this approach to higher dimensions remains unclear. The authors intend to tackle this issue in future work. Secondly, it is implied in [gromov-four-lectures, Section 5.5] that the same construction is expected to carry over to degenerate bands over the torus, provided that the warping function is log-concave. However, [wang2023scalar, Example 4.1] shows that this condition is not sufficient for the rigidity of torical bands and, a fortiori, for the existence of a $\mu$-bubble with the desired properties. The authors also plan to investigate general conditions for the existence of $\mu$-bubbles in open incomplete bands in future work.

Instead of focusing on degenerate spherical bands, this paper adopts a different strategy to establish A, as outlined below.

1. (1)

   We first rule out from A the case when all the inequalities are strict, following Gromov’s approach. This involves utilizing $\mu$-bubbles and Theorem 1.2 in dimension three. More precisely, we make use of these techniques to prove a comparison theorem with scalar and mean curvature bounds for compact spherical bands. Our comparison results are in the same spirit as [cecchini2022scalar, Raede23].

2. (2)

   We then employ the harmonic map heat flow coupled with the Ricci flow to demonstrate that the general case of A reduces to the situation where all the inequalities are strict, unless the metric $g$ is Einstein with $\operatorname{Ric}_{g}=3g$. Here, we make use of recent results of Lee and Tam [LeeTam], showing that the harmonic map heat flow coupled with the Ricci flow provides appropriate control of the Lipschitz constant with respect to the change of the scalar curvature under Ricci flow.

3. (3)

   Finally, we prove A for Einstein manifolds, which follows as a consequence of Bishop’s volume comparison theorem.

This approach offers a novel perspective on proving A, circumventing the difficulties associated with degenerate spherical bands and exploiting powerful tools from geometric analysis.

The remainder of this paper is organized as follows. Section 2 reviews relevant results on the existence and properties of $\mu$-bubbles on compact Riemannian bands. In Section 3, we prove a comparison result for compact spherical bands with scalar and mean curvature bound. In Section 4, we use the harmonic map heat flow coupled with the Ricci flow to show that Theorem A can be reduced to the case where all inequalities become strict unless the metric $g$ is Einstein. Finally, in Section 5 we show that the comparison theorem from Section 3 suffices to rule out strict inequalities from A and that A holds for Einstein metrics. This completes the proof of A.

We review the properties of $\mu$-bubbles to be utilized in this paper. For more details, we refer the reader to the work of Gromov [gromov-four-lectures] and Zhu [zhu_width_mu].

Let us start by establishing notation and conventions. Let $(X,g)$ be an oriented Riemannian manifold. We stress that, throughout this paper, all manifolds are oriented and connected. We denote the Ricci curvature tensor by $\operatorname{Ric}_{g}$, and the scalar curvature by $\operatorname{Sc}_{g}$. The Riemannian volume form is denoted by $dV_{g}$. For an embedded hypersurface $Z\subset X$, $g_{Z}$ stands for the restriction of $g$ to $Z$. If the boundary $\partial X$ of $X$ is non-empty, $\operatorname{A}_{g}(\partial X)$ denotes the second fundamental form with respect to the inward unit normal field $\nu$ along $\partial X$, and $\operatorname{H}_{g}(\partial X)$ its trace, the mean curvature with respect to $\nu$. As per our convention, the boundary of the closed unit ball in $\operatorname{\mathbb{R}}^{n}$ has mean curvature equal to $n-1$.

A band is a compact connected manifold with boundary $V$ together with a decomposition $\partial V=\partial_{-}V\sqcup\partial_{+}V$, where $\partial_{\pm}V$ are non-empty unions of components. A proper separating hypersurface for a band $V$ is a closed embedded hypersurface $\Sigma\subset{\kern 0.0ptV}^{\mathrm{o}}$ such that no connected component of $V\setminus\Sigma$ contains a path $\gamma\colon[0,1]\to V$ with $\gamma(0)\in\partial_{-}V$ and $\gamma(1)\in\partial_{+}V$. Note that a proper separating hypersurface $\Sigma$ for an orientable band $V$ is also orientable. Additionally, if $(V,g)$ is a Riemannian band and $\Sigma$ is a proper separating hypersurface for $V$, we denote by $(\hat{V},g)$ the Riemannian band isometrically embedded in $(V,g)$ such that $\partial_{-}\hat{V}=\partial_{-}V$ and $\partial_{+}\hat{V}=\Sigma$. We adopt the convention of denoting by $\operatorname{A}_{g}(\Sigma)$ and $\operatorname{H}_{g}(\Sigma)$ respectively the second fundamental form and the mean curvature on $\Sigma$ regarded as a union of components of $\partial\hat{V}$.

Let us now turn to $\mu$-bubbles. Let $(V,g)$ be an $n$-dimensional Riemannian band. We denote by $\mathcal{C}(V)$ the set of all Caccioppoli sets $\hat{\Omega}$ in $V$ such that $\hat{\Omega}$ contains an open neighborhood of $\partial_{-}V$ and $\hat{\Omega}\cap\partial_{+}V=\emptyset$. For the notion of Caccioppoli sets and their properties, we refer the reader to [giusti-minimal-surfaces]. If $\hat{\Omega}$ is a smooth Caccioppoli set in $\mathcal{C}(V)$, then $\partial\hat{\Omega}\cap{\kern 0.0ptV}^{\mathrm{o}}$ is a separating hypersurface for $V$. For any given smooth function $\mu$ on $V$, we consider the functional

where $\mathcal{H}^{k}$ denotes the $k$-dimensional Hausdorff measure on $(V,g)$, and $\partial^{\ast}\hat{\Omega}$ denotes the reduced boundary of $\hat{\Omega}$. We say that a Caccioppoli set $\Omega\in\mathcal{C}(V)$ is a $\mu$-bubble if it minimizes the functional $\mathcal{A}_{\mu}$, that is, if $\Omega$ satisfies

We prove a comparison theorem for spherical bands with scalar and mean curvature bounds. Let $I=[t_{-},t_{+}]$ be a compact interval, and let $\phi\colon I\to\operatorname{\mathbb{R}}_{+}$ be a smooth function. Consider the four-dimensional band $\mathbb{S}^{3}_{I}\coloneqq\mathbb{S}^{3}\times I$, equipped with the warped product metric

Let $h_{\phi}\colon I\to\operatorname{\mathbb{R}}$ be the smooth function defined as

Utilizing the classical formula for the scalar curvature of warped product metrics [spingeometry, Ch.IV, Formula (6.15)],

Additionally, $\partial_{\pm}\mathbb{S}^{3}_{I}=\mathbb{S}_{3}\times\{t_{\pm}\}$ and

We say that a positive smooth function $\phi(t)$ is log-concave if $\log(\phi(t))$ is a concave function.

To prove this proposition, we will construct a separating hypersurface for $V$ with suitable properties forcing the degree of $f$ to be zero. Let us first introduce some additional notation. Let $\operatorname{pr}_{\mathbb{S}^{3}}\colon\mathbb{S}^{3}_{I}=\mathbb{S}^{3}\times I\to\mathbb{S}^{3}$ be the projection onto the first factor. Under the hypotheses of Proposition 3.1, let $Z\subset V$ be a closed hypersurface. Consider the smooth map

where $i_{Z}\colon Z\hookrightarrow V$ denotes the inclusion of $Z$ into $V$. The next lemma contains well-known properties of the map $f_{Z}$, see [Raede23, Lemma 6.3]. We include a brief proof for clarity.

For a smooth map $f\colon(X,g)\to(Y,h)$ of Riemannian manifolds, we denote by $\left\|df_{p}\right\|$ the operator norm of the linear map $df_{p}\colon(T_{p}X,g_{p})\to(T_{f(p)}Y,h_{p})$, that is, $\left\|df_{p}\right\|$ is the infimum of the numbers $c>0$ such that $h_{f(p)}(df_{p}v,df_{p}v)^{1/2}\leq cg_{p}(v,v)^{1/2}$, for all $v\in T_{p}X$. We denote by $\left\|df\right\|\colon X\to\operatorname{\mathbb{R}}_{\geq 0}$ the function whose value at a point $p\in X$ is $\left\|df_{p}\right\|$. Note that $f$ is distance non-increasing if and only if $\left\|df_{p}\right\|\leq 1$ for all $p\in X$.

Under the hypotheses of Proposition 3.1, let $Z\subset V$ be a closed hypersurface. Consider the second-order formally self-adjoint elliptic differential operator

where $\Delta_{g_{Z}}$ denotes the Laplace-Beltrami operator of $(Z,g_{Z})$ and $f_{Z}\colon Z\to\mathbb{S}^{3}$ is defined by (3.4). Note that we adopt the sign convention for $\Delta_{g_{Z}}$ such that

With this convention, $-\Delta_{g_{Z}}$ has nonnegative spectrum. By classical elliptic theory, the spectrum of $\mathcal{L}_{g_{Z}}$ is a discrete set bounded from below consisting only of eigenvalues. Moreover, the eigenfunctions relative to each eigenvalue are smooth.

To prove Proposition 3.1, let us specialize Theorem 1.2 to the three-dimensional case. Since all three-dimensional oriented manifolds are spin, we obtain:

We are now ready to prove Proposition 3.1. We proceed in a similar way as in [schoen-yau-psc-manifolds]. Under the hypotheses of Proposition 3.1, we use the spectral information from Lemma 3.4 to make a conformal change of the metric on (a suitable component of) the three-dimensional $\mu$-bubble in such a way that the new metric would contradict Proposition 3.5 if $f$ had non-zero degree.

We employ estimates by Lee and Tam [LeeTam] on the harmonic map heat flow coupled with the Ricci flow to demonstrate that, under the hypotheses of A, if the metric $g$ is non-Einstein, then there exists a metric $\tilde{g}$ and a function $\tilde{f}$ satisfying the hypotheses of A with all inequalities being strict. For a comprehensive overview of the Ricci flow and the harmonic map heat flow, we recommend referring to [topping-lectures-ricci-flow] and [lin-wang-harmonic-maps], respectively. Recall that a metric $g$ is Einstein if $\operatorname{Ric}_{g}=cg$ for some constant $c$, known as the proportionality constant of $g$. For a smooth map $f\colon(M,g)\to(N,h)$ of Riemannian manifolds, we denote by $\operatorname{Lip}(f)$ the Lipschitz constant of $f$.

Recall that a Ricci flow on a manifold $M$ is a smooth family of Riemannian metrics $(g_{t})_{t\in[0,T]}$ on $M$ satisfying the Ricci equation

The short-time existence and primary properties of the Ricci flow were established by Hamilton in his seminal work [hamilton-three-manifolds]. The following lemma summarizes well-known properties of the Ricci flow utilized in this paper. We include a brief proof for clarity.

By utilizing estimates of Lee and Tam [LeeTam] for the harmonic map heat flow coupled with the Ricci flow, we derive the following lemma.

We combine the results from Sections 3 and 4 to establish the rigidity properties stated in A. First, we utilize Proposition 3.1 to address the case where all the inequalities in A are strict.

Before presenting the proof, we introduce some additional notation. For $\ell\in(0,\pi/2)$, consider the spherical band

Let $p_{\pm}$ be antipodal points in $\mathbb{S}^{4}$. Utilizing the identification $(\mathbb{S}^{4}\setminus\{p_{\pm}\},g_{\mathbb{S}^{4}})\cong(\mathbb{S}^{3}\times(-\pi/2,\pi/2),\cos^{2}(t)g_{\mathbb{S}^{3}}+dt^{2})$, we view $(\mathbb{S}^{3}_{\ell},g_{\cos})$ as a Riemannian band isometrically embedded in $(\mathbb{S}^{4}\setminus\{p_{\pm}\},g_{\mathbb{S}^{4}})$.