Hodge theory on ALG$^*$ manifolds

TL;DR

This paper develops a Fredholm theory for the Hodge Laplacian on ALG* manifolds, leading to new results on harmonic functions, cohomology, and gravitational instantons in four dimensions.

Contribution

It introduces a Fredholm framework for the Hodge Laplacian on ALG* manifolds and applies it to harmonic functions, cohomology, and gravitational instantons.

Findings

Existence of harmonic functions with prescribed asymptotics.

Vanishing of the first Betti number under non-negative Ricci curvature.

Determination of the optimal order of ALG* gravitational instantons.

Abstract

We develop a Fredholm Theory for the Hodge Laplacian in weighted spaces on ALG$^*$ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG$^*$ manifolds with non-negative Ricci curvature having group $\Gamma = \{e\}$ at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG$^*$ manifold. A corollary of this is vanishing of the first betti number for any ALG$^*$ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG$^*$ gravitational instantons.