Curvature, Cones, and Characteristic Numbers

TL;DR

This paper investigates Einstein metrics on 4-manifolds with edge-cone singularities, deriving modified topological formulas that reveal obstructions and provide insights into gravitational instantons as limits of these singular spaces.

Contribution

It introduces modified Gauss-Bonnet and signature theorems for 4-manifolds with edge-cone singularities, leading to new obstructions for Einstein metrics and applications to gravitational instantons.

Findings

Derived new integral formulas for edge-cone singularities.

Identified obstructions to Einstein metrics in this setting.

Gained insights into gravitational instantons as limits.

Abstract

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss-Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulae to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.