2014/06/13 by Anda Degeratu, Rafe Mazzeo · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Conical surface #Elliptic operator #Fredholm integral equation #Fredholm theory #Geometric Analysis and Curvature Flows #Geometry #Integral equation #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Pure mathematics #math.AP #math.DG
paper · pdf · doi:10.1112/plms.12105
arxiv created 2014/06/13 · openalex publication_date 2017/12/29 · arxiv updated 2018/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We consider the mapping properties of generalized Laplace-type operators L = ∇ ∗ ∇ + R on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE manifolds considered by Joyce. Our main result gives conditions under which such operators are Fredholm when acting between certain weighted Sobolev or weighted Hölder spaces. These are generalizations of well-known theorems in the asymptotically conical (or asymptotically Euclidean) setting, and also sharpen and extend corresponding theorems by Joyce. The methods here are based on heat kernel estimates originating from old ideas of Moser and Nash, as developed further by Grigor'yan and Saloff-Coste. As demonstrated by Joyce's work, the QAC spaces here contain many examples of gravitational instantons, and this work is motivated by various applications to manifolds with special holonomy.