Obata-Type Rigidity on Static Manifolds with Boundary

TL;DR

This paper proves a rigidity theorem for static metrics on manifolds with boundary, showing conditions under which the curvature map is locally surjective, contrasting with the boundaryless case.

Contribution

It establishes an Obata-type rigidity theorem for static metrics on manifolds with boundary and identifies new geometric conditions for local surjectivity of the curvature map.

Findings

Static metrics on manifolds with boundary can have locally surjective curvature maps.

The rigidity theorem characterizes when static metrics are uniquely determined by curvature data.

Boundary conditions influence the deformability of static metrics, unlike in boundaryless cases.

Abstract

We investigate static metrics on simple manifolds with compact boundary and establish an Obata-type rigidity theorem. We identify new sufficient geometric conditions under which the combined curvature map $g\mapsto (R_g, H_g)$ is a local surjection. Consequently, we demonstrate that in contrast to manifolds without boundary, where staticity obstructs deformability, the scalar curvature map can be locally surjective at static metrics on manifolds with boundary.