A compactness theorem for surfaces with Bounded Integral Curvature

TL;DR

This paper establishes a compactness theorem for metrics with bounded integral curvature on closed surfaces, enabling a better understanding of the space of Riemannian metrics with conical singularities, including cases with accumulating singularities.

Contribution

It introduces a new compactness theorem for metrics with bounded integral curvature, extending the understanding of Riemannian metrics with conical singularities and their compactification.

Findings

Proves a compactness theorem for metrics with bounded integral curvature.

Provides a compactification of the space of Riemannian metrics with conical singularities.

Allows for accumulation of singularities in the compactification.

Abstract

We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation of singularities is allowed.