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Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature

Hongyi Sheng

TL;DR

The paper investigates non-generic domains (static manifolds with boundary) as the obstructions to local surjectivity of the scalar curvature and boundary mean curvature map. By introducing the adjoint operator $L^*$ and the boundary-compatibility operator $\Phi^*$, it derives structural properties (constant $R$ and $H$, umbilic boundary, and rigidity features) and establishes sharp dimension bounds for static potentials. It then classifies simple non-generic domains in space forms $\mathbb{R}^n$, $\mathbb{S}^n$, and $\mathbb{H}^n$, as well as in the Schwarzschild manifold, identifying explicit static potentials and kernel spaces for each case and connecting these to rigidity theorems and photon-sphere geometry. The results extend to domains with multiple boundary components via intersections and reveal deep links to geometric inequalities, the positive mass theorem, and gravitational lensing phenomena. Overall, the work clarifies how non-genericity constrains geometry and facilitates rigidity results in both mathematical relativity and differential geometry contexts.

Abstract

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. The non-generic case (also called non-generic domains) corresponds to static manifolds with boundary. We discuss their geometric properties, which also work as the necessary conditions of non-generic metrics. In space forms and the Schwarzschild manifold, we classify simple non-generic domains (with only one boundary component) and show their connection with rigidity theorems and the Schwarzschild photon sphere.

Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature

TL;DR

The paper investigates non-generic domains (static manifolds with boundary) as the obstructions to local surjectivity of the scalar curvature and boundary mean curvature map. By introducing the adjoint operator and the boundary-compatibility operator , it derives structural properties (constant and , umbilic boundary, and rigidity features) and establishes sharp dimension bounds for static potentials. It then classifies simple non-generic domains in space forms , , and , as well as in the Schwarzschild manifold, identifying explicit static potentials and kernel spaces for each case and connecting these to rigidity theorems and photon-sphere geometry. The results extend to domains with multiple boundary components via intersections and reveal deep links to geometric inequalities, the positive mass theorem, and gravitational lensing phenomena. Overall, the work clarifies how non-genericity constrains geometry and facilitates rigidity results in both mathematical relativity and differential geometry contexts.

Abstract

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. The non-generic case (also called non-generic domains) corresponds to static manifolds with boundary. We discuss their geometric properties, which also work as the necessary conditions of non-generic metrics. In space forms and the Schwarzschild manifold, we classify simple non-generic domains (with only one boundary component) and show their connection with rigidity theorems and the Schwarzschild photon sphere.
Paper Structure (9 sections, 29 theorems, 69 equations, 2 figures, 1 table)

This paper contains 9 sections, 29 theorems, 69 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compact Non-generic Domains
  4. Non-generic Domains in Space Forms and in the Schwarzschild Manifold
  5. The Euclidean space $\mathbb{R}^n$
  6. The unit sphere $\mathbb{S}^n$
  7. The hyperbolic space $\mathbb{H}^n$
  8. The Schwarzschild manifold
  9. Non-generic domains with multiple boundary components

Key Result

Proposition 1.1

Let $(M,g)$ be a Riemannian manifold. Then $u$ is a static potential if and only if the warped product metric $\bar{g}\equiv-u^2 \, dt^2+g$ is Einstein (away from the zeros of $u$).

Figures (2)

  • Figure 1: A hypersphere in the half-space model, where $O$ is its Euclidean center
  • Figure 2: The Schwarzschild photon sphere and the horizon

Theorems & Definitions (48)

  • Proposition 1.1: Corvino C
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Corollary 2.6
  • proof
  • Proposition 3.1: Cruz-Vitório
  • Proposition 3.2
  • ...and 38 more