The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds

TL;DR

This paper extends the classical Chern-Gauss-Bonnet formula to certain 4-dimensional manifolds with singularities and conformally flat ends, incorporating isoperimetric deficits and broadening geometric understanding.

Contribution

It generalizes the Chern-Gauss-Bonnet formula to 4D manifolds with singularities, including isolated branch points and conical singularities, under specific curvature conditions.

Findings

Derived a new Chern-Gauss-Bonnet formula with error terms as isoperimetric deficits.

Extended previous smooth case results to manifolds with singular points.

First such formula in higher dimensions accommodating singularities.

Abstract

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a new Chern-Gauss-Bonnet formula with error terms that can be expressed as isoperimetric deficits. This is the first such formula in a dimension higher than two which allows the underlying manifold to have isolated branch points or conical singularities.