Desingularizing isolated conical singularities: Uniform estimates via weighted Sobolev spaces

We define a very general “parametric connect sum” construction that 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) Poincaré 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 [15] concerning desingularizations of special Lagrangian conifolds in $\mathbb{C}^m$.

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Published 9 April 2013