On the volume functional of compact manifolds with boundary with constant scalar curvature

Abstract

Abstract. We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and "small" hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

Brown–York mass and positive scalar curvature II: Besse’s conjecture and related problems

“…Here we adopt the definition in [9] instead of the original one in [18] (where µ is taken to be 1) and explain the reason later.…”

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

“…where Ric g and Hess g f stand, respectively, for the Ricci tensor and the Hessian of f associated to g on M n (see [23] for more details). Hence, following the terminology used in [2,8,9,24] we recall the definition of Miao-Tam critical metric.…”

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

“…Theorem 4.21 in [9]), and this result stimulated several interesting works. In this spirit, Miao and Tam [20,21] studied variational properties of the volume functional constrained to the space of metrics of constant scalar curvature on a given compact manifold with boundary. Indeed, volume is one of the natural geometric quantities used to study geometrical and topological properties of a Riemannian manifold.…”

“…Let (M n , g) be a connected compact Riemannian manifold with dimension n at least three and smooth boundary ∂M. According to [5,7,14,20] and [21], we say that g is, for simplicity, a Miao-Tam critical metric if there is a nonnegative smooth function f on M n such that f −1 (0) = ∂M and satisfies the overdetermined-elliptic system…”

“…It is proved in [20,21] that these critical metrics arise as critical points of the volume functional on M n when restricted to the class of metrics g with prescribed constant scalar curvature such that g | T ∂M = h for a prescribed Riemannian metric h on the boundary. Corvino, Eichmair and Miao [14] studied the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant.…”