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$M$上的一个有界Lipschitz域$\Sigma$，我们证明了$$\mu_{3-\beta_1} \leq \lambda_{1},$$，其中$\mu_k$（分别地$\lambda_k$）是第$k$个Neumann（Dirichlet）问题。 曲面上的拉普拉斯特征值在$\Sigma$和$\beta_1$上分别是$\Sigma.$的第一贝蒂数。如果$\Sigma$光滑且单连通，则可以进一步推导出严格的不等式$ \mu_{3}< \lambda_{1}. $。这将欧几里得空间上的先前结果推广到了各种曲面，包括平坦圆柱、双曲平面、双曲尖点、颈圈、漏斗以及像悬链面和螺旋面这样的极小曲面。 本文的新颖之处在于通过曲面上$1$形式的霍奇拉普拉斯算子的变分原理，比较了狄利克雷和诺依曼拉普拉斯算子的特征值，扩展了 Rohleder 在欧几里得平面上针对向量场开发的变分原理。 这种比较被简化为存在具有适当曲率条件的水平集距离函数的问题。 显示英文摘要

摘要： For a bounded Lipschitz domain $\Sigma$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that $$\mu_{3-\beta_1} \leq \lambda_{1},$$ where $\mu_k$ ($\lambda_k$ resp.) is the $k$-th Neumann (Dirichlet resp.) Laplacian eigenvalue on $\Sigma$ and $\beta_1$ is the first Betti number of $\Sigma.$ If $\Sigma$ is smooth and simply connected, we can further derive the strict inequality $ \mu_{3}< \lambda_{1}. $ This extends previous results on the Euclidean space to various curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, collar, funnel, and minimal surfaces such as catenoid and helicoid. 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.