Static manifolds with boundary: Their geometry and some uniqueness theorems

TL;DR

This work studies static manifolds with boundary $(M,g,V)$ satisfying $\operatorname{Hess}_gV-(\Delta_gV)g-V\operatorname{Ric}_g=0$, linking the geometry and topology of $M$ to the zero-level set $\Sigma=V^{-1}(0)$. It develops a coherent program to understand how $\Sigma$ controls boundary behavior, via index theory for minimal/free-boundary and constant-mean-curvature surfaces, together with spectral tools (Laplacian and Steklov). In the compact setting with nonpositive cosmological constant, it proves connectivity and intersection results for $\Sigma$, derives a Shen–type integral identity for $|\mathring{\operatorname{Ric}_g}|$, and establishes isoperimetric/rigidity conclusions, including that the round Euclidean ball is uniquely characterized under a Morse-index-one condition on $\Sigma$. In the noncompact case, the exterior of photon spheres in Schwarzschild and AdS–Schwarzschild geometries are classified as static manifolds with boundary, reinforcing the link between photon-sphere geometry and static manifolds. An appendix completes the picture by classifying conformally flat static domains within canonical model spaces.

Abstract

Static manifolds with boundary were recently introduced to mathematics. This kind of manifold appears naturally in the prescribed scalar curvature problem on manifolds with boundary when the mean curvature of the boundary is also prescribed. They are also interesting from the point of view of general relativity. For example, the (time-slice of the) photon sphere on the Riemannian Schwarzschild manifold splits it into static manifolds with boundary. In this paper, we prove a number of theorems that relate the topology and geometry of a given static manifold with boundary to some properties of the zero-level set of its potential (such as connectedness and closedness). Also, we characterize the round ball in the Euclidean 3-space with standard potential as the only scalar-flat static manifold with mean-convex boundary whose zero-level set of the potential has Morse index one. This result follows from a general isoperimetric inequality for 3-dimensional static manifolds with boundary, whose zero-level set of the potential has Morse index one. Finally, we prove some uniqueness theorems for the domains bounded by the photon sphere on the Riemannian Schwarzschild manifold.

Theorems & Definitions (58)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Remark 1.10