Log-Concavity of the First Eigenfunction on Horoconvex Domains in the Hyperbolic Plane

Let $\mathrm{\Omega }\subset {\mathbb{H}}^2$ be a bounded smooth horoconvex domain and let ${\psi }_1>0$ be its first Dirichlet eigenfunction. We prove that $${\mathrm{Hess}}_{{\mathbb{H}}^2}(-\log {\psi }_1)>0$$ throughout $\mathrm{\Omega }$, 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.