Hodge theory on ALG^∗ manifolds

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 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.

Gravitational instantons with quadratic volume growth

“…In this subsection, we review the Gibbons–Hawking construction of the $${\rm ALG}^*$$ model metric. See [10, Section 2] for more details. Let $$\nu$$ be any positive integer.…”

“…Proof Since $$(r e^{\sqrt {-1}\theta _{1}})^2$$ is invariant under $$\iota$$, by [10, Theorem 1.3], there exists a harmonic function $$z$$ on $$X$$ such that $$\Phi ^* z$$ is asymptotic to $$(r e^{\sqrt {-1}\theta _{1}})^2$$ on $$\mathfrak {M}_{2\nu }(R)$$. Since $$$$\begin{equation} \operatorname{I}_{\mathfrak {M}}^* d ((r e^{\sqrt {-1}\theta _{1}})^2) = \sqrt {-1}d ((r e^{\sqrt {-1}\theta _{1}})^2), \end{equation}$$$$we have $$$$\begin{equation} \eta \equiv \operatorname{I}_{X}^* d z - \sqrt {-1}d z = O(s^{-1 + \mu }) \end{equation}$$$$for all …”

“…By Subsection 2.2, the hyperkähler structure is $$\operatorname{ALG}^*$$ according to Definition 1.3. Finally, the order can be improved to 2 by [10, Theorem 1.10].…”

“…Remark We note that $$\operatorname{ALG}^*$$ gravitational instantons satisfy the following properties. (1)$$\operatorname{Vol}(B_t(x_0)) \sim t^2$$ as $$t \rightarrow \infty$$. (2)The tangent cone at infinity is $$\mathbb {R}^2/ \lbrace \pm 1\rbrace$$. (3)$$|\operatorname{Rm}_g| = O(s^{-2} (\log s)^{-1})$$ as $$s \rightarrow \infty$$. (4)There exist $$\operatorname{ALG}^*$$ coordinates on $$X$$ so that the order satisfies $$\mathfrak {n} \geqslant 2$$; see [10, Theorem 1.10]. …”

“…We first show that the elliptic surface constructed in Proposition 3.1 is a rational elliptic surface. To see this, it follows from [10,Corollary 1.6] that the first betti number 𝑏 1 (𝑋) = 0. Since 𝐷 is a configuration of 2-spheres corresponding to an extended Dynkin diagram of dihedral type, we have 𝑏 1 (𝐷) = 0.…”