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We study the volume functional on the space of constant scalar curvaturemetrics with a prescribed boundary metric. We derive a sufficient and necessarycondition for a metric to be a critical point, and show that the only domainsin space forms, on which the standard metrics are critical points, are geodesicballs. In the zero scalar curvature case, assuming the boundary can beisometrically embedded in the Euclidean space as a compact strictly convexhypersurface, we show that the volume of a critical point is always no lessthan 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 toa standard Euclidean ball. We also derive a second variation formula and applyit to show that, on Euclidean balls and ''small'' hyperbolic and sphericalballs in dimensions 3 to 5, the standard space form metrics are indeed saddlepoints for the volume functional.

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