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
Γ
=
{
e
}
\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.