Inserted: 15 sep 2011
We define a very general ``parametric connect sum'' construction which can be used to eliminate isolated conical singularities on 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.
For a geometric application of these results we refer the reader to our paper ``Special Lagrangian conifolds, II'' concerning desingularizations of special Lagrangian conifolds in complex space $C^m$.