Spherical conical metrics and harmonic maps to spheres-猫眼课题宝

DOI：

10.1090/tran/8578

发表时间：

2021-04

期刊：

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其他

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作者：

M. Karpukhin;Xuwen Zhu

曲面$\Sigma$上的球锥度量$g$是常曲率度量$1$，具有许多孤立的锥奇点。当至少一个锥角超过$2\pi$时，这种度量的均匀化问题在很大程度上仍然是开放的。本征函数的弗里德里希拉普拉斯$\Delta_g$与本征值$\lambda=2$在这个问题中发挥了特殊的作用，因为它们代表当地的障碍变形的度量$g$类的球锥度量。在本文中，我们将球面多值调和映射理论应用于此类本征函数的存在性问题。在第一部分中，我们建立了一个新的准则的存在性$2$-本征函数，给出了一定的亚纯数据$\Sigma$。作为应用，我们给出了球面上最多有三个圆锥奇点的度量的全部2 $-本征函数的描述。第二部分是利用多值调和映射的变形构造具有大量2 $-本征函数的度量。我们提供了新的明确的例子，通过这两种方法的度量与许多$2$-本征函数，并描述了一般的算法来找到度量与任意大的$2$-本征函数。

A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds $2\pi$. The eigenfunctions of the Friedrichs Laplacian $\Delta_g$ with eigenvalue $\lambda=2$ play a special role in this problem, as they represent local obstructions to deformations of the metric $g$ in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of $2$-eigenfunctions, given in terms of a certain meromorphic data on $\Sigma$. As an application we give a description of all $2$-eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of $2$-eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many $2$-eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of $2$-eigenfunctions.

Spherical conical metrics and harmonic maps to spheres

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