Super Log-concavity of the First Eigenfunctions for Horo-convex Domains in Hyperbolic Space

TL;DR

The paper proves that the first Dirichlet eigenfunction of the Laplacian on horo-convex domains in hyperbolic space $\mathbb{H}^n$ is super log-concave when the diameter is not too large, with optimality demonstrated by counterexamples in non-horo-convex or large-diameter cases. The authors analyze geodesic balls, proving strict $1$-log-concavity for small radii, and employ a curvature-flow deformation that preserves horo-convexity and converges to a geodesic sphere. By coupling this deformation with a maximum-principle-based argument on the Hessian of the log-eigenfunction, they show that strict $1$-log-concavity holds along the flow and passes to the limit, yielding super log-concavity for the original horo-convex domain. The results yield new log-concavity estimates in negatively curved ambient spaces and open avenues for spectral-gap bounds via one-dimensional model comparisons, with potential implications for Dirichlet-Neumann gap estimates on such domains.

Abstract

In this paper, we prove that the first eigenfunction of the Laplacian for a horo-convex domain $Ω\subset\mathbb H^n$ is super log-concave when $\text{diam}(Ω)$ is not large. Our result is optimal in the sense that there are counterexamples %are constructed for the cases when $Ω$ is not horo-convex or when $\text{diam}(Ω)$ is large respectively

Figures (1)

• Figure 1: Graph of $\Phi"-\Phi'$

Theorems & Definitions (12)

• Definition 1
• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Theorem 3.1: Theorem 1.6 of HLW22
• Proposition 3.2: Proposition 6.1 of HLW22
• Definition 2
• Lemma 4.1
• proof