Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 167-181

PRESCRIBING GAUSSIAN CURVATURE ON CLOSED RIEMANN SURFACE WITH CONICAL SINGULARITY IN THE NEGATIVE CASE

Yunyan Yang and Xiaobao Zhu

Renmin University of China, Department of Mathematics
Beijing 100872, P.R. China; yunyanyang 'at' ruc.edu.cn

Renmin University of China, Department of Mathematics
Beijing 100872, P.R. China; zhuxiaobao 'at' ruc.edu.cn

Abstract. The problem of prescribing Gaussian curvature on Riemann surface with conical singularities is considered. Let (Σ,β) be a closed Riemann surface with a divisor β, and K_λ = K + λ, where K ; Σ → R is a Hölder continuous function satisfying max_ΣK = 0, K ≠ 0, and λ ∈ R. If the Euler characteristic χ(Σ,β) is negative, then by a variational method, it is proved that there exists a constant λ^* > 0 such that for any λ ≤ 0, there is a unique conformal metric with the Gaussian curvature K_λ; for any λ, 0 < λ < λ^*, there are at least two conformal metrics having K_λ its Gaussian curvature; for λ = λ^*, there is at least one conformal metric with the Gaussian curvature K_λ*; for any λ > λ^*, there is no certain conformal metric having Ksub>λ its Gaussian curvature. This result is an analog of that of Ding and Liu [16], partly resembles that of Borer, Galimberti and Struwe [5], and generalizes that of Troyanov [28] in the negative case.

2010 Mathematics Subject Classification: Primary 58E30, 53C20.

Key words: Prescribing Gaussian curvature, conical singularity.

Reference to this article: Y. Yang and X. Zhu: Prescribing Gaussian curvature on closed Riemann surface with conical singularity in the negative case. Ann. Acad. Sci. Fenn. Math. 44 (2019), 167-181.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4411