Desingularizing isolated conical singularities: Uniform estimates via weighted Sobolev spaces

TL;DR

This paper introduces a general method to smooth out isolated conical singularities in Riemannian manifolds using parametric connect sums, establishing uniform analytic estimates in weighted Sobolev spaces.

Contribution

It develops a broad parametric connect sum construction for desingularization and proves uniform elliptic estimates independent of parameters in weighted Sobolev spaces.

Findings

Uniform Sobolev embedding estimates established

Invertibility of Laplace operator shown to be uniform

Poincare and Gagliardo-Nirenberg-Sobolev inequalities proven

Abstract

We define a very general "parametric connect sum" construction which can be used to eliminate isolated conical singularities of Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used in the construction. Specifically, we prove uniform estimates related to (i) Sobolev Embedding Theorems, (ii) the invertibility of the Laplace operator and (iii) Poincare' and Gagliardo-Nirenberg-Sobolev type inequalities. Our main tools are the well-known theories of weighted Sobolev spaces and elliptic operators on "conifolds". We provide an overview of both, together with an extension of the former to general Riemannian manifolds. For a geometric application of our results we refer the reader to our paper "Special Lagrangian conifolds, II: Gluing constructions in C^m".