Existence and nonexistence results for a nonlocal isoperimetric problem on ℍⁿ

2025/12/14 by Haizhong Li, Bo Yang, Li, Haizhong +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Contact Mechanics and Variational Inequalities

paper · pdf · doi:10.48550/arxiv.2512.12621

Abstract

In Euclidean space ℝⁿ, 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 ℍⁿ, 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.

Citations

• On minimizers in the liquid drop model
• On the existence and boundedness of minimizing measures for a general form of non-local energies
• Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion
• Existence of minimizers for a generalized liquid drop model with fractional perimeter
• Existence and nonexistence in the liquid drop model
• Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds
• Minimality of balls in the small volume regime for a general Gamow type functional
• Minimisers of a general Riesz-type Problem
• A nonlocal isoperimetric problem with density perimeter
• Anisotropic liquid drop models
• A compactness lemma and its application to the existence of minimizers for the liquid drop model
• Existence and stability for a non-local isoperimetric model of charged liquid drops
• Small Volume Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional

Related

• Non-Local Isoperimetric Problems
• Remark on a nonlocal isoperimetric problem
• Isoperimetric problems for a nonlocal perimeter of Minkowski type
• Approaching the isoperimetric problem in H^m_ℂ via the hyperbolic log-convex density conjecture
• A local isoperimetric inequality for balls with nonpositive curvature