论文标题
在表面上的拉普拉斯特征值的共形最大指标
Conformally maximal metrics for Laplace eigenvalues on surfaces
论文作者
论文摘要
该论文涉及在给定体积的表面上用riemannian度量标准的拉普拉斯特征值的最大化。 Nadirashvili-sire和Petrides使用相关的方法不同，但使用不同的方法来实现了这一问题的重大进展。尤其是，对于给定的$ k $，在表面上的共形类中的$ k $ th拉普拉斯特征值的最大值要么在标准上获得平滑的度量，除非在有限数量的圆锥形奇异点处获得，要么在表面上形成“泡泡树”，而在极限上获得了极限。从几何上讲，在这种情况下出现的气泡树可以看作是触摸相同圆形球体的结合。我们提供了这一说法的另一个证明，开发了第二作者和Y. Sire提出的方法。作为侧面结果，我们在表面的拓扑光谱上提供明确的上限。
The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods. In particular, it was shown that for a given $k$, the maximum of the $k$-th Laplace eigenvalue in a conformal class on a surface is either attained on a metric which is smooth except possibly at a finite number of conical singularities, or it is attained in the limit while a "bubble tree" is formed on a surface. Geometrically, the bubble tree appearing in this setting can be viewed as a union of touching identical round spheres. We present another proof of this statement, developing the approach proposed by the second author and Y. Sire. As a side result, we provide explicit upper bounds on the topological spectrum of surfaces.
