The stationary points of the total scalar curvature functional on the space
of unit volume metrics on a given closed manifold are known to be precisely the
Einstein metrics. One may consider 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. In this paper, we localize a condition satisfied
by such stationary points to smooth bounded domains. The condition involves a
generalization of the static equations, and we interpret solutions (and their
boundary values) of this equation variationally. On domains carrying a metric
that does not satisfy the condition, we establish a local deformation theorem
that allows one to achieve simultaneously small prescribed changes of the
scalar curvature and of the volume by a compactly supported variation of the
metric. We apply this result to obtain a localized gluing theorem for constant
scalar curvature metrics in which the total volume is preserved. Finally, we
note that starting from a counterexample of Min-Oo's conjecture such as that of
Brendle-Marques-Neves, counterexamples of arbitrarily large volume and
different topological types can be constructed.Comment: All comments welcome! Published version: Math. Ann. (to appear
