Gauss-Bonnet formula for metrics with logarithmic singularities on compact Riemann surfaces

2025/08/03 by Yuanjiu Lyu, Bin Xu, Lyu, Yuanjiu +1
Mathematics · #14H15 #14H15 53C21 #53C21 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2508.01632

openalex publication_date 2025/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a generalization of the classical Gauss-Bonnet formula for metrics with logarithmic singularities on compact Riemann surfaces, under the condition that the Gaussian curvature is Lebesgue integrable with respect to the metric's area form. As an application, we establish the Gauss-Bonnet formula for special Kähler metrics when their associated cubic differentials are meromorphic on compact Riemann surfaces.

Citations

• The Gauss-Bonnet formula of a conical metric on a compact Riemann surface
• Special Kähler structures, cubic differentials and hyperbolic metrics

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