Log-Concavity and Level-Set Horoconvexity of the First Eigenfunction on Horoconvex Domains in the Hyperbolic Plane

Abstract

Let $\Omega \subset {\mathbb{H}}^2$ be a bounded smooth horoconvex domain and let ${\psi }_1>0$ be its first Dirichlet eigenfunction. We prove that \[ \operatorname{Hess}_{\mathbb H^2}(-\logψ_1)>0 \] throughout $\Omega$, with no restriction on the diameter or the first eigenvalue. The proof is by contradiction. A degenerate Hessian would yield a shifted translation Killing derivative with a singular interior zero. Then the boundary-zero theorem of Grossi and Provenzano shows that the shifted Killing derivative has exactly two zeros on the boundary. A nodal-domain argument on the surface rules this out. As an application we prove that every superlevel set of ${\psi }_1$ is horoconvex: every level curve has geodesic curvature at least $1$. The Hessian bound makes the shifted construction available for Killing fields with nonvanishing rotation part, and yields the pointwise inequality $|(\mathrm{Hess}u{)}^{-1}J\nabla u|\le 1$ for $u=-\log {\psi }_1$, where $J$ is rotation by $\pi /2$; a boundary-zero count for translation fields with arbitrary axis completes the argument.

arXiv comment14 pages, comments are welcome