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Curvature inequalities and rigidity for constant mean curvature and spacetime constant mean curvature surfaces

Alejandro Peñuela Diaz

Abstract

We establish curvature inequalities and rigidity results for surfaces satisfying constant mean curvature type conditions in both Riemannian and Lorentzian geometry. In the Riemannian setting we study constant mean curvature (CMC) surfaces in three-dimensional manifolds with scalar curvature bounds. Building on the Christodoulou-Yau inequality $H^2\leq 16π/ |Σ|$ (with $H$ the mean curvature and $|Σ|$ the area), we show that the associated rigidity phenomena persist under a weaker notion of stability controlling only the constant mode of the second variation, combined with an extrinsic curvature sign condition. This yields Euclidean rigidity without imposing intrinsic symmetry or near-roundness assumptions and extends to higher dimensions and to the hyperbolic and spherical settings. In the Lorentzian setting we consider spacetime constant mean curvature (STCMC) surfaces, a natural generalization of CMC surfaces. We introduce a stability theory for STCMC surfaces and prove the sharp inequality $|\vec{H}|^2\leq 16π/ |Σ|$ under the dominant energy condition. We also obtain rigidity for the equality case: under suitable geometric assumptions the surface is intrinsically round and the spacetime region it bounds is flat, with maximal globally hyperbolic development isometric to a causal diamond in Minkowski spacetime. Finally, we show that the canonical asymptotic STCMC foliations known in both the spacelike and null settings have leaves that are stable with respect to this notion of stability.

Curvature inequalities and rigidity for constant mean curvature and spacetime constant mean curvature surfaces

Abstract

We establish curvature inequalities and rigidity results for surfaces satisfying constant mean curvature type conditions in both Riemannian and Lorentzian geometry. In the Riemannian setting we study constant mean curvature (CMC) surfaces in three-dimensional manifolds with scalar curvature bounds. Building on the Christodoulou-Yau inequality (with the mean curvature and the area), we show that the associated rigidity phenomena persist under a weaker notion of stability controlling only the constant mode of the second variation, combined with an extrinsic curvature sign condition. This yields Euclidean rigidity without imposing intrinsic symmetry or near-roundness assumptions and extends to higher dimensions and to the hyperbolic and spherical settings. In the Lorentzian setting we consider spacetime constant mean curvature (STCMC) surfaces, a natural generalization of CMC surfaces. We introduce a stability theory for STCMC surfaces and prove the sharp inequality under the dominant energy condition. We also obtain rigidity for the equality case: under suitable geometric assumptions the surface is intrinsically round and the spacetime region it bounds is flat, with maximal globally hyperbolic development isometric to a causal diamond in Minkowski spacetime. Finally, we show that the canonical asymptotic STCMC foliations known in both the spacelike and null settings have leaves that are stable with respect to this notion of stability.
Paper Structure (19 sections, 22 theorems, 204 equations, 1 figure)

This paper contains 19 sections, 22 theorems, 204 equations, 1 figure.

Table of Contents

  1. Introduction and Results
  2. Main results
  3. Constant mean curvature surfaces in Riemannian manifolds with scalar curvature bounds
  4. The Euclidean case:
  5. The hyperbolic case:
  6. The spherical case:
  7. Geometry of spacelike surfaces and STCMC surfaces
  8. Spacelike surfaces and null geometry
  9. Spacetime constant mean curvature surfaces
  10. Second variation of area
  11. Stability of STCMC surfaces
  12. Curvature inequalities and rigidity for stable STCMC surfaces
  13. Maximal globally hyperbolic developments
  14. Examples and stability on STCMC foliations
  15. Examples of stable STCMC surfaces
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $(M,g)$ be a $3$-dimensional Riemannian manifold with nonnegative scalar curvature. If $\Sigma\subset M$ is a stable constant mean curvature surface, then Here, stability is understood in the usual volume-preserving sense recalled above.

Figures (1)

  • Figure 1: Schematic picture of the causal diamond $D(\Omega)$ in Minkowski spacetime. The surface $\Sigma$ is the edge of the diamond, and the shaded region is a spacelike filling.

Theorems & Definitions (46)

  • Theorem 2.1: Christodoulou-Yau Chriyau
  • Theorem 2.2: shi2019uniquenesssun2017rigidity
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 36 more