The fundamental gap of horoconvex domains in $\mathbb H^n$

TL;DR

This paper investigates the behavior of the fundamental gap in horoconvex domains within hyperbolic space, revealing that the gap times the squared diameter lacks a positive lower bound, and improves eigenvalue bounds for hyperbolic balls.

Contribution

It demonstrates that the fundamental gap times the squared diameter is unbounded in horoconvex domains and provides improved bounds for eigenvalues of hyperbolic balls.

Findings

The product of the fundamental gap and the squared diameter has no positive lower bound.

The fundamental gap of geodesic balls tends to zero as radius increases.

An improved lower bound for the first eigenvalue of hyperbolic balls is established.

Abstract

We show that, for horoconvex domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. The result follows from the study of the fundamental gap of geodesic balls as the radius goes to infinity. In the process, we improve the lower bound for the first eigenvalue of balls in hyperbolic space.