On the Schlafli differential formula

TL;DR

This paper generalizes Schlafli's differential formula to Einstein manifolds, relating volume variations to mean curvature, and applies it to extend classical inequalities and rigidity results in differential geometry.

Contribution

It introduces a smooth analogue of Schlafli's formula for Einstein manifolds, extending classical polyhedral results to smooth hypersurfaces in curved spaces.

Findings

Derived a general volume variation formula involving mean curvature in Einstein manifolds.

Extended Euclidean inequalities to constant curvature spaces.

Proved rigidity results for Ricci-flat manifolds with umbilic boundaries.

Abstract

he celebrated formula of Schlafli relates the variation of the dihedral angles of a smooth family of polyhedra in a space form and the variation of volume. We give a smooth analogue of this classical formula -- our result relates the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. The argument is direct, and the classical polyhedral result (as well as results for Lorenzian space forms) is an easy corollary. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary. We apply our results to extend the classical Euclidean inequalities of Aleksandrov to other 3-dimensional constant curvature spaces. We also obtain rigidity results for Ricci-flat manifolds with umbilic boundaries and existence results for foliations of Einstein manifolds by hypersurfaces.