Rigidity of spherical product Ricci solitons

The question of rigidity - whether a geometric object is isolated in its class, modulo the symmetries of that class - has been studied for various geometric structures in many contexts. For Ricci solitons, low dimensional solitons have been essentially classified, so their rigidity or non-rigidity is clear (the dimension $2$ case is due to Hamilton [Ham88] and the dimension $3$ case is due to Ivey [Ive93]). In higher dimensions, the classification is far from well-understood. From a geometric PDE perspective, Ricci solitons are natural generalizations of Einstein manifolds. Kröncke studied the local rigidity of Ricci solitons in [Krö16], using similar techniques as for Einstein manifolds in [Krö15b].

For non-rigidity results, similar to the Einstein case, Inoue [Ino19] studied the moduli space of Fano manifolds with Kähler-Ricci solitons. As a consequence, Inoue proved that some Kähler-Ricci solitons are non-rigid. It seems that the local rigidity of a Ricci soliton actually has deep connections to the topological/algebraic structure of the manifold, and we look forward to such interpretations for the class of Ricci solitons we study in this paper.

The study of rigidity of Einstein manifolds has a more developed history. The first rigidity result for Einstein manifolds was proved by Berger in [Ber66], where Berger proved that all Einstein metrics on $S^{n}$ whose sectional curvature is $(n-2)(n-1)$-pinched are isometric to the standard sphere metric. We consider this to be a ‘global’ result in that it gives a large explicit neighbourhood in which the special structure is unique. Later, Berger-Ebin [BE69] proposed a general strategy to prove local rigidity of Einstein metrics. In particular, they introduced the notion of “infinitesimal deformations”. The local rigidity of symmetric spaces as Einstein manifolds was first studied by [Koi80], and Kröncke extended Koiso’s result in [Krö15b].

Meanwhile, it is known that certain moduli spaces of Einstein manifolds, even with an extra structure (e.g. Kähler), can be nontrivial. Then any metric in such a moduli space is an example of non-rigid Einstein manifold. For example, even in dimension 2, the moduli space of surfaces with genus $g$ of constant curvature $-1$ has dimension $6g-6$. Hence when $g>1$, such hyperbolic metrics are not rigid. We refer the readers to [Bes87, Chapter 12.J] for further discussion.

The Ricci soliton equation is both more general and more complicated than the Einstein equation, making it more challenging to obtain a rigidity result. It is also interesting to compare Ricci flow with other geometric flows, especially the most natural extrinsic flow - the mean curvature flow. There are a number of results on the rigidity of shrinking solitons to the mean curvature flow - see [CIM15], [ELS20], [SZ20], [Zhu20]. All these results are about product self-shrinkers. During the completion of this work, we learnt that Colding-Minicozzi are developing techniques which could be applied to study certain noncompact Ricci solitons [CMa, CMb].

The starting point in the study of local rigidity is to consider infinitesimal deformations which satisfy the equation $\Phi=0$ to first order. That is, one looks for $2$-tensors in the kernel of the linearized operator $\mathcal{D}\Phi$. By analogy with the mean curvature flow setting, we call elements of this kernel Jacobi deformations; in the Ricci flow literature these are sometimes referred to as infinitesimal solitonic deformations [Krö16]. A Jacobi deformation is called integrable if it induces a continuous family of gradient shrinking Ricci solitons. Any diffeomorphism or rescaling generates an integrable Jacobi deformation.

In [Krö16], Kröncke found conditions under which certain Einstein manifolds are rigid as gradient shrinking Ricci solitons. His idea was to expand $\Phi=0$ to higher and higher order and examine whether there are solutions to each order. If there are no solutions at a given order, we call this an obstruction of that order. Kröncke calculates only up to order 2: $\mathbb{CP}^{2n}$ has a second order obstruction, whilst for $S^{2}\times S^{2}$ all Jacobi deformations are integrable to second order. (For $\mathbb{CP}^{2n+1}$, almost all Jacobi deformations are non-integrable to second order, but there is a set of positive codimension which is indeed integrable.)

The manifolds we study in this paper, also have Jacobi deformations which are integrable to order 2, but we will show they have an obstruction at order 3. In particular, we resolve the case of $S^{2}\times S^{2}$. We also hope that our method may be adapted to determine the rigidity of $\mathbb{CP}^{2n+1}$. As described below, expanding at orders higher than 2 result in inhomogenous elliptic PDEs, which makes the analysis significantly more complicated. For instance, we use the special structure of eigenfunctions on $S^{2}$ to solve (1.6) explicitly.

In [SZ20], we studied the rigidity of products of spheres $S^{m}\times S^{n}$ as mean curvature flow self-shrinkers, and found that despite the existence of nontrivial Jacobi deformations, there is an obstruction at third order. (Note that [ELS20], by a similar method, had previously covered the $m,n=1$ case.) As mentioned above, when $m,n\geq 3$, as a Ricci shrinker $S^{m}\times S^{n}$ does not have Jacobi deformations other than those from symmetry (a first order obstruction). However, when at least one of the factors is $S^{2}$, nontrivial Jacobi deformations appear once more. We remark that when considered in the class of Einstein manifolds, there are again no nontrivial deformations, even for $S^{2}$ products.

Our proof strategy is a deformation-obstruction theory for the shrinker quantity $\Phi$, mirroring that we used in [SZ20] (also developed by the second named author in [Zhu20]). One may consult those papers for an overview of the method, including in toy cases, but we reproduce a sketch below for the readers’ convenience. The basic idea is similar to that of Kröncke [Krö16], but we study the expansion of $\Phi$ more directly to find a ‘uniformly’ obstructed term, which gives the quantitative rigidity.

Consider an elliptic functional $\mathcal{F}$ defined on a Banach space $\mathcal{E}$. Let $\mathcal{G}:\mathcal{E}\to\mathcal{E}^{\prime}$ be the Euler-Lagrange operator of $\mathcal{F}$ and suppose $0$ is a critical point of $\mathcal{F}$, so that $\mathcal{G}(0)=0$. In the Ricci soliton setting, $\mathcal{F}$ will be the shrinker entropy $\nu$ and $\mathcal{G}$ will be the Ricci shrinker quantity $\Phi(g+\cdot)$.

For $h\in\mathcal{E}$, we have the formal Taylor expansion

Here $\mathcal{D}\mathcal{G}(h)=Lh$, where $L$ is the linearised operator at 0. Its kernel $\mathcal{K}$ is the space of Jacobi deformations.

If $\mathcal{K}=0$, ellipticity gives that $L$ is invertible, hence $\mathcal{G}$ is obstructed at order 1. (As mentioned above, modulo symmetries, this is the case for $S^{m}\times S^{n}$, $m,n\geq 3$.) Otherwise $\mathcal{K}\neq 0$, and the first order term suggests the decomposition $h=h_{1}+k_{1}$, where $k_{1}\in\mathcal{K}$ and $h_{1}$ is in a complement $\mathcal{E}_{1}$ of $\mathcal{K}$. This gives the further expansion

Invertibility of $L$ on $\mathcal{E}_{1}$ implies that $h_{1}$ is (at least) second order compared to $h$ and $k_{1}$. The second order term is then $Lh_{1}+\frac{1}{2}\mathcal{D}^{2}\mathcal{G}(k_{1},k_{1})$. By elliptic theory, there exists $h_{1}$ for which this term is 0, if and only if $\pi_{\mathcal{K}}(\mathcal{D}^{2}\mathcal{G}(k_{1},k_{1}))=0$. In [Krö16], Kröncke essentially finds Einstein manifolds for which $\pi_{\mathcal{K}}(\mathcal{D}^{2}\mathcal{G}(k_{1},k_{1}))\neq 0$ for all nonzero $k_{1}\in\mathcal{K}$. This would imply $\|\pi_{\mathcal{K}}(\mathcal{D}^{2}\mathcal{G}(k_{1},k_{1}))\|\geq\delta\|k_{1}\|^{2}$, which we refer to as an order 2 obstruction, and in turn implies $\|h\|\simeq\|k_{1}\|\lesssim\|\mathcal{G}\|^{\frac{1}{2}}$.

For the cases considered in this paper, it turns out that $\pi_{\mathcal{K}}(\mathcal{D}^{2}\mathcal{G}(k_{1},k_{1}))\equiv 0$. That is, there is always a solution of

for some $k_{2}\in\mathcal{E}_{1}$, which suggests the further decomposition $h_{1}=k_{2}+h_{2}$. The expansion then becomes

where again invertibility of $L$ shows that the residual $h_{2}$ is of third order compared to $h$. Repeating the process, we may attempt to solve the third order condition

which depends on the projection $\pi_{\mathcal{K}}\left(\mathcal{D}^{2}\mathcal{G}(k_{1},h_{2})+\frac{1}{6}\mathcal{D}^{3}\mathcal{G}(k_{1},k_{1},k_{1})\right).$ Note that the third order condition involves some lower order cross terms.

If the projection vanishes identically, we proceed to the next higher order. If at the $m$th order, the corresponding projection is nonzero for any nonzero $k_{1}\in\mathcal{K}$, then we have an $m$th order obstruction, which would imply the quantitative rigidity $\|h\|\lesssim\|\mathcal{G}\|^{\frac{1}{m}}$. Again, for the cases considered here, we find a third order obstruction

We collect some preliminary results in Section 2 and record descriptions of the Jacobi deformations on our $S^{2}$ products in Section 3. In Section 4 we compute the variation of the shrinker quantities $\nu,\tau,f,\Phi$, with the variation formulae for more basic geometric quantities recorded in Appendix A. Using the explicit description of Jacobi deformations, we specialise these variations to our $S^{2}$ products in Section 5 and establish a formal obstruction at order 3. Finally, in Section 6, we use Taylor expansion and the formal obstruction to prove the main quantitative rigidity Theorem 1.3. The a priori bounds on $\mathcal{D}^{k}\Phi$ are deferred to Appendix B.

Consider a soliton metric $\Phi(g)=0$. We consider the formal expansion of $\Phi$ given by

where $\mathcal{D}^{k}\Phi_{g}$ is a symmetric $k$-linear map $\mathcal{S}^{2}(M)^{k}\to\mathcal{S}^{2}(M)$. For instance, $L=\mathcal{D}\Phi_{g}$ is the linearisation of $\Phi$ at $g$. We will henceforth suppress dependence on the initial metric $g$. We will use the corresponding notation for variations of other quantities such as $\nu(g),\tau_{g},f_{g}$.

Given a 1-parameter family of metrics $g(s)$ with $g(0)=g$, the corresponding infinitesimal deformation is $g^{\prime}(0)\in\mathcal{S}^{2}(M)$. At a shrinking soliton $\Phi(g)=0$, we define $\mathcal{K}:=\ker L$ to be the kernel of $L=\mathcal{D}\Phi_{g}$. We call elements of $\mathcal{K}$ Jacobi deformations; they correspond to deformations which preserve $\Phi=0$ to first order and have previously been referred to as infinitesimal solitonic deformations.

The Ricci shrinker quantity has two natural symmetries: under scaling by $c>0$ we have $\Phi(cg)=c\Phi(g)$, and under pullback by a diffeomorphism $\psi\in\operatorname*{Diff}(M)$ we have $\Phi(\psi^{*}g)=\psi^{*}\Phi(g)$. The corresponding spaces of infinitesimal deformations are $\mathbb{R}g$ and $\operatorname*{im}\mathcal{L}$, and if $\Phi(g)=0$ these are integrable subspaces of Jacobi deformations. We define the subspace of Jacobi deformations from symmetry to be $\mathcal{K}_{0}:=\operatorname*{im}\mathcal{L}+\mathbb{R}g\subset\mathcal{K}$.

If $g$ is a $\mu$-Einstein metric, Kröncke [Krö16, Section 6] checked that

Note that $\operatorname{Hess}^{g}v=\frac{1}{2}\mathcal{L}(\nabla v)^{\sharp}$. A theorem of Lichnerowicz ([Lic58], see also [IV15, Theorem 2.1]) implies that $\lambda_{1}(\Delta)>\mu$, and in particular that $\Delta+\mu$ is invertible.

Let $\mathring{C}^{\infty}_{g}=\{u\in C^{\infty}(M)|\int v\mathop{}\!\mathrm{d}V_{g}=0\}$, so that the conformal deformation space satisfies

Without loss of generality, we will assume $\mu=1$. The action of $L$ is well-known and also computed in Section 4.1: It acts on the conformal part $vg$, where $v=(\Delta+1)\tilde{v}$, $\tilde{v}\in\mathring{C}^{\infty}_{g}$, by

and on the TT part $h\in\mathring{\mathcal{S}}^{2}_{g}$ by

Recall that we defined $\Delta_{L}$ as an operator on $\mathring{\mathcal{S}}^{2}_{g}$. The above implies the $L^{2}_{g}$-orthogonal decomposition of the kernel

where

We denote by $\pi_{\mathcal{K}}$ the $L^{2}$-projection to $\mathcal{K}$, and similarly for other subspaces.

To study deformations of a gradient shrinking Ricci soliton, we need to account for the symmetry action at an integrated level, not just infinitesimal. More precisely, given a gradient shrinking Ricci soliton $(M,g)$ and a metric $\tilde{g}$ close to $g$, we want to compare $\tilde{g}$ to the symmetry orbit of $g$ (or vice versa).

The proof is standard, using implicit function theorem for Banach spaces. We refer the readers to the proof of Theorem 3.6 in [Via14].

By a standard separation of variables, spectra on a product manifold can be derived from the spectra of each component. For the Laplacian on functions, we have

For the spectrum of $\Delta_{L}$, it is convenient to first work on all of $\mathcal{S}^{2}(M)$. To match with previous literature, we define the Einstein operator $\Delta_{E}:\mathcal{S}^{2}(M)\to\mathcal{S}^{2}(M)$ by

Note that this differs from the expression for $\Delta_{L}$ by exactly the Ricci terms; in the case of a $\mu$-Einstein manifold, for $h\in\mathring{\mathcal{S}}^{2}_{g}$ we have $\Delta_{E}h=(\Delta_{L}+2)h$.

We have the following theorem (see [AM11] and [Krö15a]; note that our sign convention is opposite to Kröncke’s).

Moreover, the eigentensors on the product manifold may be given by (symmetric tensor) products of the corresponding eigensections.

We will always consider $S^{n}$ with the metric induced as the sphere of radius $\sqrt{n-1}$ in $\mathbb{R}^{n+1}$, which is $1$-Einstein. The spectra of the Laplace operators has been computed (see for instance [Bou99, Bou09]):

For $k,n\in\mathbb{N}$ define $\lambda^{0}_{k,n}=\frac{k(k+n-1)}{n-1}$, $\lambda^{1}_{k,n}=\frac{k(k+n-1)+n-2}{n-1}$ and $\lambda^{2}_{k,n}=\frac{k(k+n-1)+2(n-1)}{n-1}$.

• •

  The spectrum of the Laplacian on functions $\Delta_{0}$ consists of eigenfunctions $\varphi_{k,n}$ with eigenvalue $\lambda^{0}_{k,n}$, for $k\geq 0$.

• •

  The Hodge Laplacian acts on $\Omega^{1}(S^{n})$ by $\Delta_{1}-1$ and has spectrum consisting of $d\varphi_{k,n}$, $k\geq 1$, together with divergence-free eigenforms $\omega_{k,n}$ with eigenvalue $\lambda^{1}_{k,n}$, for $k\geq 1$.

• •

  The (generally defined) Lichnerowicz Laplacian acts on $\mathcal{S}^{2}(S^{n})$ by $\Delta_{E}-2$, and its spectrum consists of: $\varphi_{k,n}g$, $k\geq 0$; $\operatorname{Hess}\varphi_{k,n}$, $k\geq 2$; $\mathcal{L}\omega_{k,n}$, for $k\geq 2$; and TT eigenforms $h_{k,n}$ with eigenvalue $\lambda^{2}_{k,n}$, for $k\geq 2$.

That is,

In particular, since $\lambda^{2}_{2,n}=\frac{4n}{n-1}>2$ for any $n\geq 2$, there are no TT eigentensors of eigenvalue 0, that is, $\ker(\Delta_{L}+2)=0$.

Later we focus on the $n=2$ case (which is in fact the unit sphere), so we record separately

On $S^{2}$, the $2$-eigenspace $\ker(\Delta_{0}+2)$ consists of the coordinate functions $\theta_{i}=\langle x,e_{i}\rangle$, which satisfy $\nabla\theta_{i}=e_{i}^{T}$, $\langle\nabla\theta_{i},\nabla\theta_{j}\rangle=\delta_{ij}-\theta_{i}\theta_{j}$, $\operatorname{Hess}\theta_{i}=-\theta_{i}g$ and $\Delta_{0}(\theta_{i}\theta_{j})=2\delta_{ij}-6\theta_{i}\theta_{j}$.

If the product manifold is $S^{m}\times S^{n}$ for $m,n\geq 3$, there is no Jacobi deformation other than those induced by diffeomorphisms from the spectrum on product manifolds. This is a consequence of the following lemma.

In the following we only focus on the product manifolds with $S^{2}$ components.

We give precise descriptions of Jacobi deformations on $S^{2}\times S^{2}$ and $S^{2}\times N$. Recall that we decomposed the kernel as

where $\mathcal{K}_{0}$ is the space of Jacobi deformations from symmetries, and

Let $\pi_{1},\pi_{2}$ be the projections onto each factor.

We will consider 1-Einstein manifolds $N$, which additionally satisfy:

• ($\dagger$)

  $\lambda_{1}(\Delta^{N}_{0})>2$, and $\ker(\Delta^{N}_{L}+2)=0$.

Later, it will be convenient to collect the two previous cases as $M=\prod_{b=1}^{B}S^{2}_{b}\times\tilde{N}$. We will write any $v\in\ker(\Delta+2)$ as $v=\sum v_{b}$, where $v_{b}\in\ker(\Delta^{b}+2)$ and $\Delta^{b}$ is the Laplacian on the $b$th $S^{2}$ factor. It will be understood that each $v_{b}$ is independent of any other factors; explicitly, $v_{b}=\pi_{b}^{*}u_{b}$ in the above.

We begin with the first variation as it is distinguished in our method. So we consider the 1-parameter family of metrics $g(t)$, where to start $g^{\prime}$ is a general element of $\mathcal{S}^{2}(M)$ satisfying $\int(\operatorname{tr}_{g}g^{\prime})\mathop{}\!\mathrm{d}V_{g}=0$. Here we use primes to denote the derivative in $t$, evaluated at 0.

As $\Phi=0$ at $t=0$, we immediately have

Note that $\eta^{\prime}=(\log(w\mathop{}\!\mathrm{d}V_{g}))^{\prime}=-\frac{n\tau^{\prime}}{2\tau}-f^{\prime}+\frac{1}{2}\operatorname{tr}_{g}g^{\prime}$. Then $(\int\phi w\mathop{}\!\mathrm{d}V_{g})^{\prime}=\int w\mathop{}\!\mathrm{d}V_{g}(\phi\eta^{\prime}+\phi^{\prime}).$

Taking $\phi=(1-f_{0})+f$, since $\nu^{\prime}=0$ we have $0=\int w\mathop{}\!\mathrm{d}V_{g}(\eta^{\prime}+f^{\prime})=\int w\mathop{}\!\mathrm{d}V_{g}(-\frac{n\tau^{\prime}}{2\tau}+\frac{1}{2}\operatorname{tr}_{g}g^{\prime}).$ Therefore