双曲平面中 horoconvex 区域上第一特征函数的对数凹性
Log-Concavity and Level-Set Horoconvexity of the First Eigenfunction on Horoconvex Domains in the Hyperbolic Plane
AI总结：
针对双曲平面内有界光滑horoconvex区域的第一Dirichlet特征函数，通过反证法结合边界零点定理与结点域论证，证明其负对数的Hessian处处为正，即该特征函数满足对数凹性。
AI中文摘要：
设Ω⊂ℍ²为有界光滑horoconvex区域，ψ₁>0为其第一Dirichlet特征函数。我们证明在Ω内处处满足Hess_{ℍ²}(-logψ₁)>0，且无需对直径或第一特征值施加限制。证明采用反证法：退化的Hessian会产生带奇异内部零点的平移Killing导数，Grossi与Provenzano的边界零点定理表明该平移Killing导数在边界上恰有两个零点，而曲面上的结点域论证排除了这一情况。
英文摘要：
Let $Ω\subset\mathbb H^2$ be a bounded smooth horoconvex domain and let $ψ_1>0$ be its first Dirichlet eigenfunction. We prove that \[ \operatorname{Hess}_{\mathbb H^2}(-\logψ_1)>0 \] throughout $Ω$, 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 $ψ_1$ 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 $|(\operatorname{Hess} u)^{-1}J\nabla u|\le1$ for $u=-\logψ_1$, where $J$ is rotation by $π/2$; a boundary-zero count for translation fields with arbitrary axis completes the argument.
