Rigidity for critical metrics of the volume functional

Abstract

Geodesic balls in a simply connected space forms Sn, Rn or Hn are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. and Shen to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.

Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

“…Moreover, the equality holds if and only if M 3 is isometric to a geodesic ball in R 3 or S 3 . This result also holds for negative scalar curvature, provided that the mean curvature of the boundary satisfies H > 2, as was proven in [4]; see also [6]. Another upper bound estimate for the area of the boundary was obtained by Corvino, Eichmair and Miao (cf.…”

Geometric inequalities for critical metrics of the volume functional

“…Batista et al [6], inspired by a classical result obtained in [10] and [20], showed that the boundary ∂M of a compact three-dimensional oriented Miao-Tam critical metric (M 3 , g) with connected boundary and nonnegative scalar curvature must be a 2sphere whose area satisfies the inequality area(∂M ) ≤ 4π C , where C is a constant greater than 1. This result also holds for negative scalar curvature, provided that the mean curvature of the boundary satisfies H > 2, as was proven in [3]; see also [5]. Thereafter, Baltazar et al [2] were able to show an isoperimetric type inequality for Miao-Tam critical metrics with nonnegative scalar curvature.…”

Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

“…And then several generalizations of this rigidity result were found by different authors, replacing the Einstein assumption by a weaker condition such as harmonic Weyl tensor [3], parallel Ricci tensor [4], or cyclic parallel Ricci tensor [5]. For Some other generalizations or rigidity results, we can refer to [6][7][8][9][10], etc.…”