Spectral Gap Estimates on Conformally Flat Manifolds

TL;DR

This paper proves a conjecture relating the spectral gap of the Laplacian on horoconvex domains in hyperbolic space to the domain's diameter, using conformal log-concavity and extending to conformally flat manifolds.

Contribution

It establishes the first eigenfunction's conformal log-concavity estimates and proves the conjectured lower bounds on the spectral gap for horoconvex domains.

Findings

Proved the conjecture on spectral gap lower bounds in hyperbolic space.

Established spectral gap estimates for conformally flat manifolds.

Linked spectral gap to geometric quantities like inradius.

Abstract

The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schr\"odinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic space, Nguyen, Stancu and Wei conjectured that it is possible to obtain a lower bound on the fundamental gap in terms of the diameter of the domain and the dimension [IMRN2022]. In this article, we prove this conjecture by establishing conformal log-concavity estimates for the first eigenfunction. This builds off earlier work by the authors and Saha as well as recent work by Cho, Wei and Yang. We also prove spectral gap estimates for a more general class of problems on conformally flat manifolds and investigate the relationship between the gap and the inradius. For example, we establish gap estimates for domains in $\mathbb{S}^1 \times \mathbb{S}^{N-1}$ which are convex with respect to the universal affine cover