AI 中文总结
该研究探讨带边界流形上Dirichlet-to-Robin映射的谱变分性质，证明II型Yamabe度量使第一归一化特征值泛函取极值，还确立了第二归一化特征值最大化的广义度量的存在性及其对应方程或映射的性质。
AI 中文摘要
我们研究维数至少为3的带边界连通紧致流形上的Dirichlet-to-Robin映射$\boldsymbol{\textit{D}}_g$的谱的变分性质。对于第一特征值，我们证明II型Yamabe度量使第一归一化特征值泛函取极值，并刻画了所有极端度量；若$[g]$是使得$\boldsymbol{\textit{D}}_g$至少有两个负特征值的共形类，我们证明该共形类中存在广义度量可最大化$\boldsymbol{\textit{D}}_g$的第二归一化特征值。此外，我们证明这类度量要么定义了带边界流形上的Escobar-Yamabe型方程（在边界上变号）的解，要么是到单位欧几里得球的弱自由边界调和映射。
英文摘要
We study the variational properties of the spectrum of the Dirichlet-to-Robin map $\mathcal{D}_g$ on connected compact manifolds with boundary of dimension at least three. For the first eigenvalue, we show that Type II Yamabe metrics extremize the first normalized eigenvalue functional, and we characterize all extremals. If $[g]$ is a conformal class for which $\mathcal{D}_g$ has at least two negative eigenvalues, then we show the existence of a generalized metric that maximizes the second normalized eigenvalue of $\mathcal{D}_g$ in the conformal class. Moreover, we show that each such metric either defines a solution to an Escobar--Yamabe type equation on manifolds with boundary that changes sign along the boundary, or a weakly free-boundary harmonic map into the unit Euclidean ball.
