Conformal deformations of conic metrics to constant scalar curvature

TL;DR

This paper investigates conformal deformations of a broad class of incomplete conic metrics in higher dimensions, identifying obstructions and conditions for achieving constant scalar curvature, including existence and uniqueness results.

Contribution

It generalizes previous work by allowing more flexible links and warping, and provides new criteria for conformal deformations to constant scalar curvature in higher dimensions.

Findings

Identifies obstructions to conformal deformations to constant scalar curvature.

Provides sufficient conditions for existence of negative scalar curvature conformal metrics.

Establishes uniqueness of the negative scalar curvature conformal metric within its class.

Abstract

We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the "link" of the singular set. Within this class of "conic metrics," we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.