New Integral Estimates in Substatic Riemannian Manifolds and the Alexandrov Theorem

TL;DR

This paper develops new integral estimates for substatic Riemannian manifolds with boundary, extending classical geometric identities and inequalities, and introduces a novel vector field technique inspired by the P-function method.

Contribution

It generalizes the Alexandrov Theorem and improves the Heintze-Karcher inequality in the context of substatic manifolds using a new divergence-free vector field approach.

Findings

Derived new integral estimates for substatic manifolds

Extended Alexandrov Theorem to this setting

Enhanced Heintze-Karcher inequality

Abstract

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.