Elliptic analysis on collapsing gravitational instantons modelled using the Gibbons-Hawking ansatz

TL;DR

This paper establishes uniform elliptic regularity estimates for collapsing gravitational instantons modeled by the Gibbons-Hawking ansatz, analyzing the Laplacian's properties on these spaces.

Contribution

It provides uniform elliptic estimates and analyzes the Fredholm properties of the Laplacian on collapsing gravitational instantons.

Findings

Elliptic regularity estimates are uniform across the family of metrics.

The Laplacian is shown to be a Fredholm operator.

Identification of weighted spaces where the Laplacian is an isomorphism.

Abstract

In this paper, we will establish elliptic regularity estimates on families of gravitational instantons whose model metric near infinity collapse to a flat 3 dimensional space. We show that the constants in these estimates can be chosen uniformly for the whole family of metrics. We apply these results to the Laplacian and show it is Fredholm. We also determine between which weighted spaces it is an isomorphism.