Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

TL;DR

This paper investigates the log-concavity of the first Dirichlet eigenfunction on convex domains in positively curved surfaces, establishing new conditions for strong log-concavity and deriving bounds on the fundamental gap.

Contribution

It introduces curvature-based conditions ensuring strong log-concavity of eigenfunctions on convex domains in curved surfaces, extending previous results beyond Euclidean and spherical cases.

Findings

Strong log-concavity under curvature conditions

Lower bounds on fundamental gaps for convex domains

Behavior of eigenfunction estimates under Ricci flow

Abstract

We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first eigenfunction is strongly log-concave. Previously, for general convex domains, the log-concavity of the first eigenfunctions were only known when lying in $\mathbb{R}^n$ and $\mathbb{S}^n$. Using this estimate, we establish lower bounds on the fundamental gap of such regions. Furthermore, we study the behavior of these estimates under Ricci flow and other deformations of the metric.