标题： 曲面上狄利克雷和诺依曼特征值之间的不等式 显示英文标题

标题： Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces

摘要： 对于满足一定曲率条件的黎曼曲面上的有界Lipschitz域$Σ$和$M$，我们证明了对于任意$k \geq 1,$，有$$μ_{k+2-β_1} \leq λ_{k},$$，其中$μ_k$（分别地$λ_k$）是第$k$个Neumann（Dirichlet）问题。 曲面上的 Laplacian 特征值在$Σ$和$β_1$上是$Σ.$的第一 Betti 数。这将之前关于欧几里得空间的结果推广到了弯曲曲面，包括平坦圆柱、双曲平面、双曲尖点、漏斗等。 本文的新颖之处在于通过曲面上$1$形式的 Hodge 拉普拉斯算子的变分原理比较了 Dirichlet 和 Neumann Laplacian 特征值，扩展了 Rohleder 在欧几里得平面上对向量场发展的变分原理。 这种比较被简化为存在具有适当曲率条件的水平集的距离函数的问题。 显示英文摘要

摘要： For a bounded Lipschitz domain $Σ$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that for any $k \geq 1,$ we have $$μ_{k+2-β_1} \leq λ_{k},$$ where $μ_k$ ($λ_k$ resp.) is the $k$-th Neumann (Dirichlet resp.) Laplacian eigenvalue on $Σ$ and $β_1$ is the first Betti number of $Σ.$ This extends previous results on the Euclidean space to curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, funnel, etc. The novelty of the paper lies in comparing Dirichlet and Neumann Laplacian eigenvalues via the variational principle of the Hodge Laplacian on $1$-forms on a surface, extending the variational principle on vector fields in the Euclidean plane as developed by Rohleder. The comparison is reduced to the existence of a distance function with appropriate curvature conditions on its level sets.