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

TL;DR

This paper explores a nonlocal isoperimetric problem in hyperbolic space, proving geodesic balls are unique minimizers for small volumes and establishing nonexistence for large volumes under certain conditions.

Contribution

It extends the study of nonlocal isoperimetric problems from Euclidean to hyperbolic space, providing new existence and nonexistence results.

Findings

Geodesic balls are unique minimizers for small volumes.

Nonexistence of minimizers for large volumes under certain exponents.

Results depend on the volume size and the nonlocal term's exponent.

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.