New integral estimates in substatic manifolds and the Alexandrov Theorem

Abstract: The classical Alexandrov Theorem in the Euclidean space asserts that any bounded set with a smooth boundary of constant mean curvature is a ball. This result can be more quantitatively expressed by showing that an integral deficit from being of constant mean curvature dominates suitable analytic quantities that vanish exactly when the domain is a ball. In this talk, we provide generalizations of this in the context of substatic manifolds with boundary, that constitute a vast generalization of the family of manifolds with nonnegative Ricci curvature, and that are of particular importance in General Relativity. Our approach is based on the discovery of a vector field with nonnegative divergence involving the solution to a torsion-like boundary value problem introduced by Li-Xia in a related earlier work. The talk is based on a joint work with A. Pinamonti (Trento).

differential geometry

Audience: researchers in the topic

Differential Geometry Seminar Torino

Series comments: This is a hybrid seminar organized by the Differential Geometry groups of Università and Politecnico di Torino.

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