可约圆锥度量的谱性质
Spectral properties of reducible conical metrics
摘要：
证明了球面锥度量的单值性是可约的当且仅当它在Laplace-Beltrami算子的全纯延拓中有一个实值本征函数，其本征值为2.这样一个本征函数产生一个亚纯向量场，它与锥度规的展开映射有关。我们也给出了第一个非零特征值的下界，并根据monodromy的特征空间维数的完整分类。本文可以看作是球锥度规研究中复分析方法与偏微分方程方法之间的一种新的联系。
英文摘要：
We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace--Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of the conical metric. We also give a lower bound of the first nonzero eigenvalue, and a complete classification of the eigenspace dimension depending on the monodromy. This paper can be seen as a new connection between the complex analysis method and the PDE approach in the study of spherical conical metrics.
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