It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of metrics with conical singularities induced from branched minimal immersions by first eigenfunctions into spheres. Our primary motivation is the fact that metrics realizing maxima of the first non-zero Laplace eigenvalue are induced by minimal branched immersions into spheres. In particular, we show that the properties of such metrics induced from $\mathbb{S}^2$ differ significantly from the properties of those induced from $\mathbb{S}^m$ with $m>2$. This feature appears to be novel and needs to be taken into account in the existing proofs of the sharp upper bounds for the first non-zero eigenvalue of the Laplacian on the $2$-torus and the Klein bottle. In the present paper we address this issue and give a detailed overview of the complete proofs of these upper bounds following the works of Nadirashvili, Jakobson-Nadirashvili-Polterovich, El Soufi-Giacomini-Jazar, Nadirashvili-Sire and Petrides.

中文翻译：

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第一特征函数对表面的分支最小浸入

Montiel 和 Ros 证明，对于紧凑表面上的每个共形结构，至多有一个度量允许通过第一本征函数最小程度地浸入某个单位球体。我们将这个定理推广到具有圆锥奇点的度量设置，圆锥奇点是由第一特征函数引入球体的分支最小浸入引起的。我们的主要动机是，实现第一个非零拉普拉斯特征值最大值的度量是由最小的分支浸入球体引起的。特别是，我们表明从 $\mathbb{S}^2$ 导出的此类度量的属性与从 $\mathbb{S}^m$ 导出的属性显着不同，其中 $m>2$。这个特征看起来很新颖，需要在 $2$-torus 和 Klein 瓶上拉普拉斯算子的第一个非零特征值的锐利上限的现有证明中考虑在内。在本文中，我们解决了这个问题，并根据 Nadirashvili、Jakobson-Nadirashvili-Polterovich、El Soufi-Giacomini-Jazar、Nadirashvili-Sire 和 Petrides 的工作详细概述了这些上限的完整证明。