Communications in Analysis and Geometry

Volume 31 (2023)

Number 7

Singular hyperbolic metrics and negative subharmonic functions

Pages: 1827 – 1848

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n7.a7

Authors

Yu Feng (CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China)

Yiqian Shi (CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China)

Jijian Song (Center for Applied Mathematics, School of Mathematics, Tianjin University, Tianjin, 300350, China)

Bin Xu (CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China)

Abstract

We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is Zariski dense in $\text{PSL}(2,\,{\mathbb R})$. By using meromorphic differentials and affine connections, we obtain evidence of the conjecture that the monodromy group of the singular hyperbolic metric cannot be contained in four classes of one-dimensional Lie subgroups of $\text{PSL}(2,\,{\mathbb R})$. Moreover, we confirm the conjecture if the Riemann surface is the once punctured Riemann sphere, the twice punctured Riemann sphere, a once punctured torus or a compact Riemann surface.

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Y.F. is supported in part by China Scholarship Council. Y.S. is supported in part by the National Natural Science Foundation of China (Grant No. 11931009) and Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200). J.S. is partially supported by National Natural Science Foundation of China (Grant Nos. 12001399 and 11831013) and the International Postdoctoral Exchange Fellowship Program 2021 by the Office of China Postdoctoral Council. B.X. is supported in part by the Project of Stable Support for Youth Team in Basic Research Field, CAS (Grant No. YSBR-001) and the National Natural Science Foundation of China (Grant Nos. 11571330, 11971450 and 12071449). Both Y.S. and B.X. are supported in part by the Fundamental Research Funds for the Central Universities (Grant No. WK3470000015).

Received 25 September 2020

Accepted 2 September 2021

Published 26 July 2024