Calculus of Variations and Geometric Measure Theory

R. Buzano - H. T. Nguyen

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

created by muller on 12 Jun 2018
modified on 13 Jun 2022


Published Paper

Inserted: 12 jun 2018
Last Updated: 13 jun 2022

Journal: Communications in Analysis and Geometry
Volume: 27
Number: 8
Year: 2019
Doi: 10.4310/CAG.2019.v27.n8.a2

ArXiv: 1503.06602 PDF


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.