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

TL;DR

This work derives the first and second variation formulas for the renormalized area RenA of complete surfaces in conformally compact static Einstein spaces along a canonical flow $\partial_t\Phi = V\nu$. The analysis reveals that negativity of the Neumann data $h_3$ amplifies instability, while the renormalized-area framework yields a rigidity mechanism for warped-product torus boundaries, recovering and strengthening prior results on static AdS-like geometries. Central to the approach are precise near-boundary expansions of $\frac{1}{V^2}g$ and $\frac{1}{V}$, a graph-parametrization of minimal surfaces, and a careful cancellation of divergent terms between area and boundary length in RenA computations. The paper also establishes an existence theory for minimizers of i(Γ) via GMT, and, under a flat torus boundary, proves a rigidity statement identifying the Horowitz–Myers soliton as the unique realization under the stated conditions. Collectively, these results connect renormalized geometric quantities to stability and rigidity phenomena in conformally compact static Einstein manifolds.

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 \cite{GSW}.

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Theorems & Definitions (30)

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