Neumann Data and Second Variation Formula of Renormalized Area for Conformally Compact Static Spaces

2025/04/21 by Wang, Zhixin
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.15048

Abstract

In this paper, we derive the first and second variation formulas for the renormalized area for static Einstein spaces along a specific direction, demonstrating that the negativity of the Neumann data implies instability. Consequently, we obtain a rigidity result for the case when the conformal boundary is a warped product torus, which strengthens the result presented in \citeGSW.

Citations

• The rigid Horowitz-Myers conjecture
• On static three-manifolds with positive scalar curvature
• A note on static spaces and related problems
• Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds
• Rigidity of Conformally Compact Manifolds with the Round Sphere as the Conformal Infinity
• Rigidity of Asymptotically Hyperboblic Manifolds
• On the rigidity for conformally compact Einstein manifolds
• On the Geometry and Mass of Static, Asymptotically AdS Spacetimes, and the Uniqueness of the AdS Soliton
• The Mass of Asymptotically Hyperbolic Manifolds
• Maximum Principles for Null Hypersurfaces and Null Splitting Theorems
• Volume and Area Renormalizations for Conformally Compact Einstein Metrics
• Conformal Anomaly Of Submanifold Observables In AdS/CFT Correspondence
• The spectrum of an asymptotically hyperbolic Einstein manifold
• On the First Variation of a Varifold
• Normal and Integral Currents

Related

• Volume and Area Renormalizations for Conformally Compact Einstein Metrics
• An Inverse Problem for Renormalized Area: Determining the Bulk Metric with Minimal Surfaces
• Conformal deformations of initial data sets to the strict dominant energy condition using a spacetime Poisson equation
• On the renormalized volumes for conformally compact Einstein manifolds
• Area-charge inequalities and rigidity of time-symmetric initial data sets