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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
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