Convexity and singularities of curvature equations in conformal geometry

Details Convexity and singularities of curvature equations in conformal geometry Convexity and singularities of curvature equations in conformal geometry

2005-04-04

arxiv.org/abs/math/0504066v2

Int. Math. Res. Not. 2006, Art. ID 96890

Mathematics

Differential Geometry

35 pages; final version published in IMRN

Scientific paper

We define a generalization of convex functions, which we call $\delta$-convex functions, and show they must satisfy interior H\"older and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear equations in conformal geometry with isolated singularities, in the case of non-negative Ricci curvature. We prove that such solutions either extend to a H\"older continuous function across the singularity, or else have the same singular behavior as the fundamental solution of the conformal Laplacian. We also obtain various removable singularity theorems for these equations.

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