Communications in Analysis and Geometry
Volume 27 (2019)
The Chern–Gauss–Bonnet formula for singular non-compact four-dimensional manifolds
Pages: 1697 – 1736
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.
Parts of this work were carried out during two visits of HN at Queen Mary University of London. He would like to thank the university for its hospitality. These visits have been financially supported by RB’s Research in Pairs Grant from the London Mathematical Society as well as HN’s AK Head Travelling Scholarship from the Australian Academy of Science. RB would also like to thank the EPSRC for partially funding his research under grant number EP/M011224/1.
Received 22 February 2017
Accepted 31 August 2017
Published 21 January 2020