A functional inequality on the boundary of static manifolds

TL;DR

This paper proves a new functional inequality relating the static potential, second fundamental form, and mean curvature on the boundary of static manifolds, advancing understanding of geometric properties in static Riemannian manifolds.

Contribution

It introduces a novel functional inequality connecting key geometric quantities on the boundary of static manifolds, which was not previously established.

Findings

Establishment of a new functional inequality involving static potential and boundary geometry.

Insights into the relationship between static potential, second fundamental form, and mean curvature.

Potential applications in geometric analysis and general relativity.

Abstract

On the boundary of a compact Riemannian manifold $(\Omega, g)$ whose metric $g$ is static, we establish a functional inequality involving the static potential of $(\Omega, g)$, the second fundamental form and the mean curvature of the boundary $\partial \Omega$ respectively.