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The rigidity statement in the Horowitz-Myers conjecture

S. Brendle, P. K. Hung

TL;DR

The paper delivers an alternative proof of the Horowitz–Myers conjecture for dimensions $3 \le N \le 7$ and proves a rigidity statement: any metric achieving equality is locally isometric to a Horowitz–Myers metric. The authors implement a dimension-descending, Schoen–Yau–style strategy built on $(N,n)$-datasets with weight $\rho$ and the central condition $(\star_{N,n})$, combined with barrier methods, conformal compactification, and a stability framework for $(g,\rho)$-stationary hypersurfaces. A base case in dimension two is established, followed by an inductive descent that propagates model $(N,n)$-datasets and yields a local HM-model near infinity. The approach yields a rigorous rigidity statement for equality cases and deepens the structural understanding of asymptotically locally hyperbolic manifolds with scalar curvature bound $-N(N-1)$. Overall, the work provides a robust, geometry-driven route to the Horowitz–Myers rigidity phenomenon in low dimensions.

Abstract

In this paper, we give an alternative proof of the Horowitz-Myers conjecture in dimension $3 \leq N \leq 7$. Moreover, we show that a metric that achieves equality in the Horowitz-Myers conjecture is locally isometric to a Horowitz-Myers metric.

The rigidity statement in the Horowitz-Myers conjecture

TL;DR

The paper delivers an alternative proof of the Horowitz–Myers conjecture for dimensions and proves a rigidity statement: any metric achieving equality is locally isometric to a Horowitz–Myers metric. The authors implement a dimension-descending, Schoen–Yau–style strategy built on -datasets with weight and the central condition , combined with barrier methods, conformal compactification, and a stability framework for -stationary hypersurfaces. A base case in dimension two is established, followed by an inductive descent that propagates model -datasets and yields a local HM-model near infinity. The approach yields a rigorous rigidity statement for equality cases and deepens the structural understanding of asymptotically locally hyperbolic manifolds with scalar curvature bound . Overall, the work provides a robust, geometry-driven route to the Horowitz–Myers rigidity phenomenon in low dimensions.

Abstract

In this paper, we give an alternative proof of the Horowitz-Myers conjecture in dimension . Moreover, we show that a metric that achieves equality in the Horowitz-Myers conjecture is locally isometric to a Horowitz-Myers metric.

Paper Structure

This paper contains 12 sections, 104 theorems, 337 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Proof of property $(\star_{N,2})$ for each $N \geq 3$
  4. The case when $\kappa(x) \neq \frac{\sinh N}{1+\cosh N} + \frac{N}{\sinh N}$ for each point $x \in \Gamma_1$
  5. The case when $\kappa(x) = \frac{\sinh N}{1+\cosh N} + \frac{N}{\sinh N}$ for some point $x \in \Gamma_1$
  6. Properties of $(g,\rho)$-stationary hypersurfaces which are $(g,\rho,u)$-stable in the sense of Definition \ref{['definition.stability']}
  7. The conformal compactification and a foliation near infinity
  8. Construction of barriers
  9. Existence of $(g,\rho)$-stationary hypersurfaces which are $(g,\rho,u)$-stable in the sense of Definition \ref{['definition.stability']}
  10. Proof of Theorem \ref{['main.theorem']}
  11. Some auxiliary identities
  12. Asymptotic behavior of solutions of linear PDEs

Key Result

Theorem 1.2

Let us fix an integer $N$ with $3 \leq N \leq 7$ and a collection of positive real numbers $b_0,\hdots,b_{N-2}$. Let $\theta_0,\hdots,\theta_{N-2}$ denote the coordinate functions on $T^{N-1}$, which take values in $S^1 = \mathbb{R}/(2\pi \mathbb{Z})$. We define a flat metric $\gamma$ on $T^{N-1}$ b If the scalar curvature of $(M,g)$ is at least $-N(N-1)$, then there exists a smooth immersion $\Ps

Theorems & Definitions (121)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 111 more