September 8, 2016
TITLE: Quasi-Asymptotically Conical Geometries
ABSTRACT: In this talk we introduce the class of quasi-asymptotically conical (QAC) geometries, a less rigid Riemannian formulation of the QALE geometries introduced by Joyce in his study of crepant resolutions of Calabi-Yau orbifolds. Our set-up is in the category of real stratified spaces and Riemannian geometry. Given a QAC manifold, we identify the appropriate weighted Sobolev spaces, for which we prove the finite dimensionality of the null space for generalized Laplacians as well as their Fredholmness. We conclude with applications to metrics with special holonomy.
The methods we use are based on techniques developed in geometric analysis by Grigor’yan and Saloff-Coste, as well as Colding and Minicozzi, and Peter Li. We show that our geometries satisfy the volume doubling property and the Poincar\’e inequality, and we use these properties to analyze the heat kernel behaviour of a generalized Laplacian and to establish Li-Yau type estimates for it.
This is based on joint work with Rafe Mazzeo and work in progress with Ronan Conlon and Frederic Rochon.