  We derive new integral estimates on substatic manifolds with boundary of
horizon type, naturally arising in General Relativity. In particular, we
generalize to this setting an identity due to Magnanini-Poggesi leading to the
Alexandrov Theorem in the Euclidean space and improve on a Heintze-Karcher type
inequality due to Li-Xia. Our method relies on the introduction of a new vector
field with nonnegative divergence, generalizing to this setting the P-function
technique of Weinberger.
