Abstract
Calculus of Variations and Partial Differential Equations: Volume
36, Issue 2 (2009), Page 141 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.
