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Existence and nonexistence results for a nonlocal isoperimetric problem on $\mathbb{H}^n$

Haizhong Li, Bo Yang

TL;DR

This work extends the nonlocal isoperimetric problem to hyperbolic space $\mathbb{H}^n$ by analyzing the energy $\mathcal{E}(F)=P(F)+\gamma NL_{\alpha}(F)$ with $NL_{\alpha}$ as a Riesz-type nonlocal term. Using the upper half-space model and a $\Phi_{\lambda}$-transformation, the authors develop a hyperbolic counterpart to the Euclidean theory, including Fuglede-type estimates for near-spherical perturbations and quantitative isoperimetric controls. They prove that geodesic balls uniquely minimize $\mathcal{E}$ for small volumes (up to isometries), establish existence of minimizers in this regime, and show nonexistence of minimizers for large volumes when $0<\alpha<2$, thereby revealing a two-regime behavior governed by the volume. The results rely on a careful blend of variational techniques, regularity theory, and hyperbolic geometry, providing a solid foundation for the nonlocal isoperimetric landscape in curved spaces. These findings substantially advance understanding of how curvature influences the competition between local perimeter and nonlocal repulsion in shape optimization problems.

Abstract

In Euclidean space $\mathbb{R}^n$, the minimization problem of a nonlocal isoperimetric functional with a competition between perimeter and a nonlocal term derived from the negative power of the distance function, has been extensively studied. In this paper, we investigate this nonlocal isoperimetric problem in hyperbolic space $\mathbb{H}^n$, we prove that the geodesic balls are unique minimizers (up to hyperbolic isometries) for small volumes $m$ and obtain nonexistence results for large volumes $m$ under certain ranges of the exponent in the nonlocal term.

Existence and nonexistence results for a nonlocal isoperimetric problem on $\mathbb{H}^n$

TL;DR

This work extends the nonlocal isoperimetric problem to hyperbolic space by analyzing the energy with as a Riesz-type nonlocal term. Using the upper half-space model and a -transformation, the authors develop a hyperbolic counterpart to the Euclidean theory, including Fuglede-type estimates for near-spherical perturbations and quantitative isoperimetric controls. They prove that geodesic balls uniquely minimize for small volumes (up to isometries), establish existence of minimizers in this regime, and show nonexistence of minimizers for large volumes when , thereby revealing a two-regime behavior governed by the volume. The results rely on a careful blend of variational techniques, regularity theory, and hyperbolic geometry, providing a solid foundation for the nonlocal isoperimetric landscape in curved spaces. These findings substantially advance understanding of how curvature influences the competition between local perimeter and nonlocal repulsion in shape optimization problems.

Abstract

In Euclidean space , the minimization problem of a nonlocal isoperimetric functional with a competition between perimeter and a nonlocal term derived from the negative power of the distance function, has been extensively studied. In this paper, we investigate this nonlocal isoperimetric problem in hyperbolic space , we prove that the geodesic balls are unique minimizers (up to hyperbolic isometries) for small volumes and obtain nonexistence results for large volumes under certain ranges of the exponent in the nonlocal term.

Paper Structure

This paper contains 11 sections, 34 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper half-space model of $\mathbb{H}^n$
  4. Isoperimetric-type inequalities in $\mathbb{H}^n$
  5. Quasiminimizer of the perimeter
  6. Fuglede’s type estimates in $\mathbb{H}^n$
  7. Basic properties of minimizers
  8. Existence of minimizers
  9. Geodesic ball as the minimizer
  10. Nonexistence of minimizers
  11. The quantitive estimates in Theorem \ref{['Thm-Fuglede']}

Key Result

Theorem 1.2

For all $n\geq 2$, $0<\alpha<n$ and $\gamma>0$, there exists a positive constant $m_1$ depending on $n,\alpha$ and $\gamma$ such that for $0<m\leq m_1$, the unique minimizer (up to hyberbolic isometries) to the minimization problem Prob-mini with volume $m$ is given by a geodesic ball.

Theorems & Definitions (75)

  • Conjecture 1.1: cf. Choksi-Peletier11
  • Remark 1
  • Remark 2
  • Theorem 1.2
  • Remark 3
  • Remark 4
  • Theorem 1.3
  • Remark 5
  • Remark 6
  • Proposition 2.1
  • ...and 65 more