Log-concavity and level set curvature estimates for the first Dirichlet eigenfunction in hyperbolic space

In this paper, we study the first Dirichlet eigenfunction on bounded, smooth and strictly horo-convex domains in hyperbolic space Hⁿ. We prove its strict log-concavity in arbitrary dimension and establish the Gauss curvature bound $K>1$ for its level sets. In dimensions two and three, we further obtain quantitative lower bounds for K in terms of the minimum Gauss curvature of the boundary. In higher dimensions, the product of the two smallest principal curvatures of each level hypersurface exceeds one. Furthermore, we extend the strict log-concavity result to bounded smooth horo-convex domains. Our approach combines maximum or minimum principles for suitable auxiliary functions, boundary convexity estimates and a domain deformation argument.