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

TL;DR

This paper investigates the volume functional for constant scalar curvature metrics with fixed boundary, characterizing critical points, especially in space forms, and analyzing their stability across different geometries and dimensions.

Contribution

It provides necessary and sufficient conditions for critical points of the volume functional, identifies geodesic balls as unique critical domains in space forms, and examines stability properties in Euclidean, hyperbolic, and spherical spaces.

Findings

Critical points occur only for geodesic balls in space forms.

Volume of critical points in Euclidean space exceeds or equals the boundary volume, equality only for Euclidean balls.

Standard metrics are saddle points for the volume functional in certain dimensions.

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 to 5, the standard space form metrics are indeed saddle points for the volume functional.