June 2021 Spectral properties of reducible conical metrics
Bin Xu, Xuwen Zhu
Author Affiliations +
Illinois J. Math. 65(2): 313-337 (June 2021). DOI: 10.1215/00192082-9043431

Abstract

We show that the monodromy of a spherical conical metric g is reducible if and only if the metric g has a real-valued eigenfunction with eigenvalue 2 for the holomorphic extension ΔgHol 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 of ΔgHol, together with a complete classification of the dimension of the space of real-valued 2-eigenfunctions for ΔgHol depending on the monodromy of the metric g. 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.

Citation

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Bin Xu. Xuwen Zhu. "Spectral properties of reducible conical metrics." Illinois J. Math. 65 (2) 313 - 337, June 2021. https://doi.org/10.1215/00192082-9043431

Information

Received: 13 March 2020; Revised: 13 January 2021; Published: June 2021
First available in Project Euclid: 25 March 2021

Digital Object Identifier: 10.1215/00192082-9043431

Subjects:
Primary: 34M35
Secondary: 53C21

Rights: Copyright © 2021 by the University of Illinois at Urbana–Champaign

Vol.65 • No. 2 • June 2021
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