## Rigidity for critical metrics of the volume functional.(English)Zbl 1419.53037

Let $$(M,g)$$ be a connected compact Riemannian manifold of dimension $$m\ge3$$ with smooth connected boundary $$\Sigma$$. Let $$\sigma=g|_\Sigma$$ and let $$\mathcal{M}_\sigma^R$$ be the space of Riemannian metrics $$\tilde g$$ on $$M$$ with constant scalar curvature $$R$$ and with $$\tilde g|_\Sigma=\sigma$$. Let $$V(\tilde g)=\text{volume}(M,\tilde g)$$ be the volume functional.
In this paper, the authors present some rigidity results for the critical points of $$V:\mathcal{M}_\sigma^R\rightarrow\mathbb{R}$$. Let $$L_g^*$$ be the formal $$L^2$$ adjoint of the linearization of the scalar curvature operator, i.e., $$L_g^*(f)=-\Delta_g(f)\cdot g+\nabla_g^2f-f\text{Ric}_g$$. One says that $$(M,g,f)$$ is a Miao-Tam critical metric if $$f^{-1}(0)=\Sigma$$ and if $$L_g^*(f)=g$$. Let $$Y(S^{m-1})$$ be the Yamabe constant of the standard round sphere of dimension $$m-1$$. The authors show:
Theorem. Let $$(M,g,f)$$ be an oriented Miao-Tam critical metric where $$m\ge4$$ and $$R=m(m-1)\varepsilon$$ for $$\varepsilon\in\{0,\pm1\}$$. Assume $$\text{Ric}^\Sigma\ge R^2(m-1)^{-1}g_\Sigma$$ with $$\inf_\Sigma R^\Sigma>0$$. If $$\varepsilon=-1$$, assume that the mean curvature of $$\Sigma$$ satisfies $$H>m-1$$. Then $$|\Sigma|^{\frac2{m-1}}\le Y(S^{m-1})/\{\frac{m-2}mR+\frac{m-2}{m-1}H^2\}$$. Equality holds if and only if $$(M,g)$$ is isometric to a geodesic ball in the simply connected space form $$S^m$$, $$\mathbb{R}^m$$, or $$\mathcal{H}^m$$.
Several corollaries are derived from this result and an inequality derived for the first eigenvalue of the Jacobi operator. A rigidity result of Boucher and of Shen is extended to $$m$$-dimensional static metric that relates to the Cosmic nohair conjecture:
Corollary. The Cosmic no-hair conjecture is true when $$\Sigma$$ is isometric to a standard sphere $$S^{m-1}$$.

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C24 Rigidity results 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

### Keywords:

critical metrics; geodesic ball; space form; volume functional
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