Topological rigidity of compact manifolds supporting Sobolev-type inequalities

TL;DR

This paper proves that compact manifolds with Ricci curvature bounds supporting near-optimal Sobolev inequalities are topologically close to the sphere, and characterizes when they are isometric to the sphere based on Sobolev constants.

Contribution

It establishes topological rigidity results for manifolds supporting Sobolev inequalities close to the sphere's optimal constants, answering a question by V.H. Nguyen.

Findings

Manifolds with Ricci curvature ≥ (n-1)g supporting near-optimal Sobolev inequalities are topologically close to the sphere.

Exact optimal Sobolev constants characterize the manifold as isometric to the sphere.

The results provide a rigidity criterion linking Sobolev inequalities to manifold geometry.

Abstract

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ric$_{(M,g)}\geq (n-1)g$. If $(M,g)$ supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the standard unit sphere $(\mathbb S^n,g_0)$, we prove that $(M,g)$ is topologically close to $(\mathbb S^n,g_0)$. Moreover, the Sobolev constants on $(M,g)$ are precisely the optimal constants on the sphere $(\mathbb S^n,g_0)$ if and only if $(M,g)$ is isometric to $(\mathbb S^n,g_0)$; in particular, the latter result answers a question of V.H. Nguyen.

Equations

$${\left({\int }_M|u{|}^{\frac{2n}{n-2}}\mathrm{d}{v}_g\right)}^{\frac{n-2}{n}}\le A{\int }_M|{\nabla }_{g}u{|}^2\mathrm{d}{v}_g+B{\int }_M{u}^2\mathrm{d}{v}_{g,}\forall u\in H_1^2(M).$$

$$B\ge \frac{1}{n(n-1)}{\omega }_n^{-\frac{2}{n}}\underset{M}{\max }{\mathrm{Scal}}_{(M,g)},$$

$$A_0=\frac{4}{n(n-2)}{\omega }_n^{-\frac{2}{n}}\mathrm{and}{B}_0={\omega }_n^{-\frac{2}{n}},$$

$${\mathrm{Vol}}_g(B_M(x_0,\rho ))\ge \min {\left\{\frac{A_0}{A},\frac{B_0}{B}\right\}}^{\frac{n}{2}}{\mathrm{Vol}}_{g_0}(B_{{\mathbb{S}}^n}(y_0,\rho )).$$

$$\max \left\{\frac{A}{A_0},\frac{B}{B_0}\right\}\le (1-{\delta }_n{)}^{-\frac{2}{n}},$$

$$1\ge \frac{{\mathrm{Vol}}_g(B_M(x_0,\rho ))}{{\mathrm{Vol}}_{g_0}(B_{{\mathbb{S}}^n}(y_0,\rho ))}\ge \frac{{\mathrm{Vol}}_g(B_M(x_0,\pi ))}{{\mathrm{Vol}}_{g_0}(B_{{\mathbb{S}}^n}(y_0,\pi ))}=\frac{{\mathrm{Vol}}_g(M)}{{\mathrm{Vol}}_{g_0}({\mathbb{S}}^n)},\forall \rho \in [0,\pi ].$$

$$B\ge {\mathrm{Vol}}_g(M{)}^{-\frac{2}{n}}\ge {\mathrm{Vol}}_{g_0}({\mathbb{S}}^n{)}^{-\frac{2}{n}}={\omega }_n^{-\frac{2}{n}}=B_0.$$

$$f(\lambda )={\int }_M(\lambda -\cos d_g{)}^{2-n}\mathrm{d}{v}_g\mathrm{and}s(\lambda )={\int }_{{\mathbb{S}}^n}(\lambda -\cos d_0{)}^{2-n}\mathrm{d}{v}_{{\mathbb{S}}^n},\lambda >1,$$

$$\underset{\lambda \to 1^{+}}{\mathrm{liminf}}\frac{f(\lambda )-\lambda f^{\prime }(\lambda )}{s(\lambda )-\lambda s^{\prime }(\lambda )}\ge 1.$$

$$I(\lambda )$$

$$J(\lambda )$$

$${\mathrm{Vol}}_g(B_M(x_0,\rho ))\ge (1-\epsilon ){\tilde{\omega }}_n{\rho }^n$$

$${\mathrm{Vol}}_{g_0}(B_{{\mathbb{S}}^n}(y_0,\rho ))\le (1+\epsilon ){\tilde{\omega }}_n{\rho }^n,$$

$$\frac{I(\lambda )}{J(\lambda )}\ge \frac{(1-\epsilon )(n-2){\tilde{\omega }}_n{\int }_0^{\delta }(\lambda -\cos \rho {)}^{-n}(n\lambda -\cos \rho ){\rho }^n\sin \rho \mathrm{d}\rho }{(1+\epsilon )(n-2){\tilde{\omega }}_n{\int }_0^{\delta }(\lambda -\cos \rho {)}^{-n}(n\lambda -\cos \rho ){\rho }^n\sin \rho \mathrm{d}\rho +\tilde{s}(\lambda ,\delta ,n)},$$

$$\tilde{s}(\lambda ,\delta ,n)$$

$$\underset{\lambda \to 1}{\lim }{\int }_0^{\delta }(\lambda -\cos \rho {)}^{-n}(n\lambda -\cos \rho ){\rho }^n\sin \rho \text{d}\rho =+\infty .$$

$$\underset{\lambda \to 1}{\lim }{\int }_0^{\delta }\frac{{\rho }^{n+1}}{(\lambda -1+{\rho }^2{)}^n}\mathrm{d}\rho =+\infty .$$

$$\underset{\lambda \to 1}{\lim }\left((\lambda -1{)}^{\frac{n}{2}-1}{\int }_0^{\delta }\frac{{\rho }^{n+1}}{(\lambda -1+{\rho }^2{)}^n}\mathrm{d}\rho \right)$$

$${\left[\frac{s^{\prime \prime }(\lambda )}{(n-2)(n-1)}\right]}^{\frac{2}{2^{\ast }}}=\frac{2}{n}{\omega }_n^{-\frac{2}{n}}\left[\frac{1-{\lambda }^2}{2(n-1)}{s}^{\prime \prime }(\lambda )-\lambda s^{\prime }(\lambda )+s(\lambda )\right],\lambda >1.$$

$$H(\lambda )={\left(\frac{K_0}{C}\right)}^{\frac{n}{2}}J(s)={\left(\frac{K_0}{C}\right)}^{\frac{n}{2}}(s(\lambda )-\lambda s^{\prime }(\lambda )),$$

$${\left[-\frac{H^{\prime }(\lambda )}{\lambda (n-2)(n-1)}\right]}^{\frac{2}{2^{\ast }}}=C\left[\frac{{\lambda }^2-1}{2\lambda (n-1)}{H}^{\prime }(\lambda )+H(\lambda )\right],\lambda >1.$$

$${\left[{\int }_M(\lambda -\cos d_g{)}^{-n}\mathrm{d}{v}_g\right]}^{\frac{2}{2^{\ast }}}\le A{\int }_M(\lambda -\cos d_g{)}^{-n}{\sin }^2{d}_g\mathrm{d}{v}_g+B{\int }_M(\lambda -\cos d_g{)}^{2-n}\mathrm{d}{v}_g.$$

$${\left[\frac{f^{\prime \prime }(\lambda )}{(n-2)(n-1)}\right]}^{\frac{2}{2^{\ast }}}\le K_0\left[\frac{A}{A_0}\frac{1-{\lambda }^2}{2(n-1)}{f}^{\prime \prime }(\lambda )-\frac{A}{A_0}\lambda f^{\prime }(\lambda )+\left(\frac{2-n}{2}\frac{A}{A_0}+\frac{n}{2}\frac{B}{B_0}\right)f(\lambda )\right],$$

$$\frac{1-{\lambda }^2}{2(n-1)}{f}^{\prime \prime }(\lambda )-\lambda f^{\prime }(\lambda )+\frac{2-n}{2}f(\lambda )=\frac{n-2}{2}{\int }_M(\lambda -\cos d_g{)}^{-n}{\sin }^2{d}_g\mathrm{d}{v}_g\ge 0,$$

$${\left[\frac{f^{\prime \prime }(\lambda )}{(n-2)(n-1)}\right]}^{\frac{2}{2^{\ast }}}\le C\left[\frac{1-{\lambda }^2}{2(n-1)}{f}^{\prime \prime }(\lambda )-\lambda f^{\prime }(\lambda )+f(\lambda )\right],\lambda >1.$$

$${\left[-\frac{I^{\prime }(\lambda )}{\lambda (n-2)(n-1)}\right]}^{\frac{2}{2^{\ast }}}\le C\left[\frac{{\lambda }^2-1}{2\lambda (n-1)}{I}^{\prime }(\lambda )+I(\lambda )\right].$$

$$I(\lambda )\ge H(\lambda ),\forall \lambda >1.$$

$$\underset{\lambda \to 1^{+}}{\mathrm{liminf}}\frac{I(\lambda )}{H(\lambda )}=\underset{\lambda \to 1^{+}}{\mathrm{liminf}}\frac{f(\lambda )-\lambda f^{\prime }(\lambda )}{{\left(\frac{K_0}{C}\right)}^{\frac{n}{2}}(s(\lambda )-\lambda s^{\prime }(\lambda ))}\ge {\left(\frac{C}{K_0}\right)}^{\frac{n}{2}}>1.$$

$$I(\lambda )\ge H(\lambda ),\forall \lambda \in (1,{\delta }_0+1).$$

$${\lambda }_s:=\sup \{\lambda <{\lambda }_0:I(\lambda )=H(\lambda )\}<{\lambda }_0.$$

Full text

1. Introduction

Let $(M,g)$ be a smooth compact $n$-dimensional Riemannian manifold, $n\geq 3.$ The general theory of Sobolev inequalities shows that there exist $A>0$ and $B>0$ such that

[TABLE]

In fact, problem (1.1) is a part of the famous AB-program initiated by Aubin [1] concerning the optimality of the constants $A$ and $B$; for a systematic presentation of this topic, see the monograph of Hebey [5, Chapters 4 & 5]. In particular, one can prove the existence of $B>0$ such that (1.1) holds with $A=A_{0}=\frac{4}{n(n-2)}\omega_{n}^{-\frac{2}{n}}$, cf. [5, Theorem 4.6], the latter value being the optimal Talenti constant in the Sobolev embedding $H_{1}^{2}{(\mathbb{R}^{n})}\hookrightarrow L^{2^{*}}(\mathbb{R}^{n}),$ $n\geq 3,$ where $2^{*}=2n/(n-2).$ Hereafter, $\omega_{n}={\rm Vol}_{g_{0}}(\mathbb{S}^{n})$ denotes the volume of the standard unit sphere $(\mathbb{S}^{n},g_{0})$. If $u\equiv 1$ in (1.1), then we have $B\geq{\rm Vol}_{g}(M)^{-\frac{2}{n}}$, where Vol${}_{g}(S)$ denotes the volume of $S\subset M$ in $(M,g)$. Moreover, if $n\geq 4$ then the validity of (1.1) with $A=A_{0}=\frac{4}{n(n-2)}\omega_{n}^{-\frac{2}{n}}$ implies

[TABLE]

where ${\rm Scal}_{(M,g)}$ is the scalar curvature of $(M,g),$ cf. [5, Proposition 5.1].

In the model case when $(M,g)=(\mathbb{S}^{n},g_{0})$ is the standard unit sphere of $\mathbb{R}^{n+1}$, Aubin [1] proved that the optimal values of $A$ and $B$ in (1.1) are

[TABLE]

respectively; moreover, for every $\lambda>1$, the function $u_{\lambda}(x)=(\lambda-\cos d_{0}(x))^{1-\frac{2}{n}}$, $x\in\mathbb{S}^{n},$ is extremal in (1.1), see also [5, Theorem 5.1]. Hereafter, $d_{0}(x)=d_{\mathbb{S}^{n}}(y_{0},x)$, $x\in\mathbb{S}^{n},$ where $d_{\mathbb{S}^{n}}$ denotes the standard metric on $(\mathbb{S}^{n},g_{0})$ and the element $y_{0}\in\mathbb{S}^{n}$ is arbitrarily fixed. Note however that on the quotients $M=\mathbb{S}^{1}(r)\times\mathbb{S}^{2}$ of $\mathbb{S}^{3}$ endowed with its natural metric $g$ (with $r>0$ sufficiently small) inequality (1.1) is not valid for $A=A_{0}$ and $B={\rm Vol}_{g}(M)^{-\frac{2}{n}},$ see [5, Proposition 5.7].

Let $B_{M}(x,\rho)$ and $B_{\mathbb{S}^{n}}(y,\rho)$ be the open geodesic balls with radius $\rho>0$ and centers in $x\in M$ and $y\in\mathbb{S}^{n}$ in $(M,g)$ and $(\mathbb{S}^{n},g_{0})$, respectively.

Our main result reads as follows:

Theorem 1.1**.**

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ricci curvature $\mathrm{Ric}_{(M,g)}\geq(n-1)g$ and assume that the Sobolev inequality $(\ref{AB})$ holds on $(M,g)$ with some constants $A,B>0$. Then the following assertions hold:

• (i)

$A\geq A_{0}$* and $B\geq B_{0}$, where $A_{0}$ and $B_{0}$ are from $(\ref{AnullBnull});$*

• (ii)

there exists $x_{0}\in M$ such that for every $y_{0}\in\mathbb{S}^{n}$ and $\rho\in[0,\pi]$,

[TABLE]

Remark 1.1**.**

Note that (1.3) is valid on the whole $[0,\infty)$. Indeed, since the Ricci curvature on $(M,g)$ verifies Ric${}_{(M,g)}\geq(n-1)g$, due to Bonnet-Myers theorem, the diameter $D_{M}:={\rm diam}(M)$ of $M$ is bounded from above by $\pi$; accordingly, for every $\rho\geq\pi$ one has $B_{M}({x_{0}},\rho)=M$ and $B_{\mathbb{S}^{n}}({y_{0}},\rho)=\mathbb{S}^{n}$, thus (1.3) can be extended beyond $\pi$.

Perelman [10] states that for every $n\geq 2$ there exists $\delta_{n}\in[0,1)$ such that if the $n$-dimensional compact Riemannian manifold $(M,g)$ with Ricci curvature $\mathrm{Ric}_{(M,g)}\geq(n-1)g$ verifies ${\rm Vol}_{g}(M)\geq(1-\delta_{n}){\rm Vol}_{g_{0}}(\mathbb{S}^{n}),$ then $M$ is homeomorphic to $\mathbb{S}^{n}$; this result has been improved by Cheeger and Colding [2, Theorem A.1.10] by replacing homeomorphic to diffeomorphic. The latter result, the equality case in Bishop-Gromov inequality and Theorem 1.1 imply the following topological rigidity for compact manifolds:

Corollary 1.1**.**

Under the same assumptions as in Theorem 1.1, if

[TABLE]

then $(M,g)$ is diffeomorphic to $(\mathbb{S}^{n},g_{0})$. Moreover, $A=A_{0}$ and $B=B_{0}$ if and only if $(M,g)$ is isometric to $(\mathbb{S}^{n},g_{0})$.

Remark 1.2**.**

The statement of Corollary 1.1 is in the spirit of the results of Ledoux [9] and do Carmo and Xia [4]. In these works certain Sobolev inequalities are considered on non-compact Riemannian manifolds with non-negative Ricci curvature, and the Riemannian manifold is isometric to the Euclidean space with the same dimension if and only if the Sobolev constants are precisely the Euclidean optimal constants. Further results in this direction can be found in the papers by Kristály [6, 7] and Kristály and Ohta [8]. Theorem 1.1 and Corollary 1.1 seem to be the first contributions within this topic in the setting of compact Riemannian manifolds, answering also a question of Nguyen [11].

2. Proofs

Proof of Theorem 1.1.

(i) The validity of the Sobolev inequality $(\ref{AB})$ on $(M,g)$ and a similar argument as in Hebey [5, Proposition 4.2] imply that $A\geq A_{0}$.

By Remark 1.1, we have $D_{M}:=$diam$(M)\leq\pi$. Since Ric${}_{(M,g)}\geq(n-1)g$, by the Bishop-Gromov comparison principle we have that for every $x_{0}\in M$ and $y_{0}\in\mathbb{S}^{n}$, the function $\rho\mapsto\frac{{\rm Vol}_{g}(B_{M}({x_{0}},\rho))}{{\rm Vol}_{g_{0}}(B_{\mathbb{S}^{n}}({y_{0}},\rho))}$ is non-increasing on $(0,\infty)$; in particular, we have

[TABLE]

Now, choosing $u\equiv 1$ in (1.1), it follows that

[TABLE]

(ii) If $D_{M}=\pi$, we have nothing to prove. Indeed, in this case $(M,g)$ is isometric to $(\mathbb{S}^{n},g_{0})$, see Cheng [3] and Shiohama [12], i.e., ${\rm Vol}_{g}(M)={\rm Vol}_{g_{0}}(\mathbb{S}^{n})$ and (2.1) implies at once relation (1.3).

Accordingly, we assume that $D_{M}<\pi.$ Fix $x_{0},\tilde{x}_{0}\in M$ such that $d_{g}(x_{0},\tilde{x}_{0})=D_{M},$ and $y_{0}\in\mathbb{S}^{n}$. Let ${\rm d}v_{g}$ and ${\rm d}v_{g_{0}}$ be the canonical volume forms on $(M,g)$ and $(\mathbb{S}^{n},g_{0})$, respectively. Let $f,s:(1,\infty)\to\mathbb{R}$ be the functions defined as

[TABLE]

where $d_{g}=d_{g}(x_{0},\cdot)$ ans $d_{0}=d_{\mathbb{S}^{n}}(y_{0},\cdot)$. It is easily seen that both functions $f$ and $s$ are well-defined and smooth on $(1,\infty)$.

The proof will be provided in several steps.

Step 1 (local behavior of $f$ and $s$ around $1$). We claim that

[TABLE]

By the layer cake representation of functions and a change of variables, we have that

[TABLE]

In a similar manner, we have

[TABLE]

Fix $\varepsilon>0$ arbitrarily. Then the local behavior of the geodesic balls both on $(M,g)$ and $(\mathbb{S}^{n},g_{0})$ implies that there exits $\delta=\delta_{\varepsilon}>0$ sufficiently small such that for every $\rho\in(0,\delta)$,

[TABLE]

and

[TABLE]

where $\tilde{\omega}_{n}$ denotes the volume of the $n$-dimensional unit ball in $\mathbb{R}^{n}$. Therefore, the above estimates give that

[TABLE]

where

[TABLE]

Note first that $\tilde{s}(\lambda,\delta,n)=O(1)$ as $\lambda\to 1.$ Now, we show that

[TABLE]

Since $\cos\rho>1-\rho^{2}$, $n\lambda-\cos\rho\geq n-1$ and $\sin\rho\geq\frac{2}{\pi}\rho$ for every $\rho\in(0,\delta)$ and $\lambda>1$, it suffices to prove that

[TABLE]

In order to check the latter limit, by changes of variables one has

[TABLE]

Step 2 (ODE vs. ODI; global comparison of $f$ and $s$). Due to Aubin [1], the extremal function in (1.1) when $(M,g)=(\mathbb{S}^{n},g_{0})$ is $u_{\lambda}(x)=(\lambda-\cos d_{0})^{1-\frac{n}{2}}$ for every $\lambda>1$. Thus, inserting $u_{\lambda}$ into (1.1) when $(M,g)=(\mathbb{S}^{n},g_{0})$ and using the notation in (2.2), we have the following ODE:

[TABLE]

Let $K_{0}=\frac{2}{n}\omega_{n}^{-\frac{2}{n}}$ and $C=K_{0}\max\left\{\frac{A}{A_{0}},\frac{B}{B_{0}}\right\}$. Without loss of generality, we may assume that $A>A_{0};$ indeed, since $A\geq A_{0}$, we may take $A=A_{0}+\varepsilon$ for $\varepsilon>0$ sufficiently small. Since $B\geq{\rm Vol}_{g}(M)^{-\frac{2}{n}}\geq B_{0}$, it turns out that $C>K_{0}.$ By introducing the function

[TABLE]

one has $H^{\prime}(\lambda)=-\lambda\left(\frac{K_{0}}{C}\right)^{\frac{n}{2}}s^{\prime\prime}(\lambda),$ therefore $s^{\prime\prime}(\lambda)=-\frac{H^{\prime}(\lambda)}{\lambda}\left(\frac{K_{0}}{C}\right)^{-\frac{n}{2}}$. This means that the second order ODE (2.6) is equivalent to the following first order ODE:

[TABLE]

Now, if we replace $w_{\lambda}(x)=(\lambda-\cos d_{g})^{1-\frac{n}{2}}$ for every $\lambda>1$ into (1.1) and we explore the eikonal equation $|\nabla_{g}d_{g}|=1$ valid a.e. on $M$, we obtain

[TABLE]

By using the notation in (2.2), the latter inequality can be rewritten into

[TABLE]

for every $\lambda>1.$ Since

[TABLE]

and $C=K_{0}\max\left\{\frac{A}{A_{0}},\frac{B}{B_{0}}\right\}$, the latter inequality implies that

[TABLE]

Since $I(\lambda)=f(\lambda)-\lambda f^{\prime}(\lambda)$, we get the following first order ordinary differential inequality:

[TABLE]

We claim that

[TABLE]

First of all, by (2.3) we clearly have that

[TABLE]

Thus, for sufficiently small $\delta_{0}>0$ one has

[TABLE]

Assume by contradiction that $I(\lambda_{0})<H(\lambda_{0})$ for some $\lambda_{0}>1$. Clearly, $\lambda_{0}>1+\delta_{0}.$ Let us define

[TABLE]

Thus for any $\lambda\in[\lambda_{s},\lambda_{0}]$ we have $I(\lambda)\leq H(\lambda).$ It is also clear that

[TABLE]

and

[TABLE]

Let us define the increasing function $\varphi_{\lambda}:(0,\infty)\to\mathbb{R}$ by

[TABLE]

By relations (2.7), (2.8) and the definition of $\varphi_{\lambda}$, for every $\lambda\in[\lambda_{s},\lambda_{0}]$ we have that

[TABLE]

Therefore, the monotonicity of $\varphi_{\lambda}$ implies

[TABLE]

In particular $\lambda\mapsto I(\lambda)-H(\lambda)$ is non-decreasing on the interval $[\lambda_{s},\lambda_{0}].$ Consequently, we have

[TABLE]

a contradiction, which shows the validity of (2.9).

Step 3 (proving (1.3)). Due to (2.1), the claim is concluded once we prove

[TABLE]

Note that relation (2.9) is equivalent to

[TABLE]

[TABLE]

for every $\lambda>1.$

Let us multiply the above inequality by $\lambda^{n-2}$ and take the limit when $\lambda\to\infty$; the Lebesgue dominance theorem implies that both integrals tend to [math], remaining

[TABLE]

Since $C=K_{0}\max\left\{\frac{A}{A_{0}},\frac{B}{B_{0}}\right\}$, the latter relation implies (2.10) at once, which concludes the proof of (1.3). ∎

Proof of Corollary 1.1.

Since $\max\left\{\frac{A}{A_{0}},\frac{B}{B_{0}}\right\}\leq(1-\delta_{n})^{-\frac{2}{n}},$ by the quantitative volume estimate (1.3) it follows that

[TABLE]

The statement follows by Cheeger and Colding [2].

If $(M,g)$ is isometric to $(\mathbb{S}^{n},g_{0})$, it is clear that $A=A_{0}$ and $B=B_{0},$ due to Aubin [1]. Conversely, when $A=A_{0}$ and $B=B_{0},$ we apply (1.3) and (2.1) in order to obtain ${\rm Vol}_{g_{0}}(B_{\mathbb{S}^{n}}(y_{0},\rho))={\rm Vol}_{g}(B_{M}(x_{0},\rho))$ for every $\rho\in[0,\pi]$ (in fact, for every $\rho\in[0,\infty))$. Now, the equality in the Bishop-Gromov comparison principle implies that $(M,g)$ is isometric to $(\mathbb{S}^{n},g_{0})$. ∎

Acknowledgment. The authors are supported by the National Research, Development and Innovation Fund of Hungary, financed under the K$\_$18 funding scheme, Project No. 127926. A. Kristály is also supported by the STAR-UBB grant.