Defining Curvature as a Measure via Gauss-Bonnet on Certain Singular Surfaces

TL;DR

This paper introduces a method to define curvature as a measure on singular surfaces formed by gluing smooth surfaces, using the Gauss-Bonnet theorem, and explores spectral properties of the Laplacian.

Contribution

It provides an explicit formula for curvature measures on singular surfaces and analyzes the spectral asymptotics of the Laplacian in this context.

Findings

Curvature measure decomposes into absolutely continuous, singular curve, and point-supported measures.

Explicit formula for curvature measure on glued singular surfaces.

Analysis of Laplacian spectral asymptotics on these surfaces.

Abstract

We show how to define curvature as a measure using the Gauss-Bonnet Theorem on a family of singular surfaces obtained by gluing together smooth surfaces along boundary curves. We find an explicit formula for the curvature measure as a sum of three types of measures: absolutely continuous measures, measures supported on singular curves, and discrete measures supported on singular points. We discuss the spectral asymptotics of the Laplacian on these surfaces.