论文标题
Min-Max谐波图和保形特征值的新特征
Min-max harmonic maps and a new characterization of conformal eigenvalues
论文作者
论文摘要
给定一个$λ_k（m，c）$的表面$ m $和固定的保形类别$ c $，是$ k $ th非平凡的laplacian特征值的至上。 Nadirashvili已经观察到，实现$λ_K（m，c）$的指标与球形的谐波图密切相关。在本文中，我们确定了$λ_1（m，c）$和$λ_2（m，c）$，具有与球体值图的能量功能相关的最小值数量。作为应用程序，我们获得了几个新的特征值边界，包括前两个steklov特征值的尖锐等等不平等。这种表征还提供了最大指标的存在的替代证明，该指标意识到$λ_1（m，c）$，$λ_2（m，c）$，此外，使我们能够获得满足自然紧凑性条件的最大ra的规则定理。
Given a surface $M$ and a fixed conformal class $c$ one defines $Λ_k(M,c)$ to be the supremum of the $k$-th nontrivial Laplacian eigenvalue over all metrics $g\in c$ of unit volume. It has been observed by Nadirashvili that the metrics achieving $Λ_k(M,c)$ are closely related to harmonic maps to spheres. In the present paper, we identify $Λ_1(M,c)$ and $Λ_2(M,c)$ with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing $Λ_1(M,c)$, $Λ_2(M,c)$ and, moreover, allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition.
