# Static potentials and area minimizing hypersurfaces

## TL;DR

This paper investigates the properties of static potentials in asymptotically flat manifolds with horizons, establishing conditions under which these potentials vanish or imply the existence of area-minimizing hypersurfaces, with implications for mass theorems.

## Contribution

It proves that static potentials vanish on the boundary and links unbounded static potentials to the existence of area-minimizing hypersurfaces, advancing understanding in geometric analysis and general relativity.

## Key findings

- Static potential must be zero on the boundary.
- Unbounded static potential implies existence of a non-compact area-minimizing hypersurface.
- Results relate to the positive mass theorem and quasi-local mass.

## Abstract

We show that if an asymptotically flat manifold with horizon boundary admits a global static potential, then the static potential must be zero on the boundary. We also show that if an asymptotically flat manifold with horizon boundary admits an unbounded static potential in the exterior region, then the manifold must contain a complete non-compact area minimizing hypersurface. Some results related to the Riemannian positive mass theorem and Bartnik's quasi-local mass are obtained.