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

TL;DR

This paper extends the Gauss-Bonnet formula to metrics with logarithmic singularities on compact Riemann surfaces, including applications to special Kähler metrics with meromorphic cubic differentials.

Contribution

It generalizes the classical Gauss-Bonnet formula to a broader class of metrics with singularities on Riemann surfaces.

Findings

Proves Gauss-Bonnet formula for metrics with logarithmic singularities.

Establishes the formula for special Kähler metrics with meromorphic cubic differentials.

Demonstrates the integrability condition for Gaussian curvature.

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 K\"ahler metrics when their associated cubic differentials are meromorphic on compact Riemann surfaces.