Fredholm theory for elliptic operators on quasi-asymptotically conical spaces

TL;DR

This paper establishes Fredholm properties of generalized Laplace operators on quasi-asymptotically conical spaces, extending classical results and providing tools for analyzing geometric operators on complex non-compact manifolds.

Contribution

It generalizes Fredholm theory for elliptic operators to QAC spaces, broadening the scope of analysis on manifolds with special holonomy and gravitational instantons.

Findings

Operators are Fredholm under specific weighted Sobolev or Hölder space conditions.

Extends and sharpens Joyce's theorems for asymptotically conical spaces.

Provides heat kernel estimates for analysis on QAC spaces.

Abstract

We consider the mapping properties of generalized Laplace-type operators ${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ 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 between certain weighted Sobolev or weighted H\"older 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.