📝 SummarXiv

Abstract

We study the natural geometric elliptic operators on a class of complete Riemannian manifolds which include the 4-dimensional ALH* gravitational instantons and their higher dimensional Calabi-Yau analogues asymptotic to the model Calabi Ansatz metrics. Some of these were initially constructed by Tian and Yau, later by Hein and most recently by Y. Chen, and we call these Tian-Yau spaces. They have played an important role in the degeneration theory of K3 metrics, cf. Hein-Sun-Viaclosky-Zhang and Sun-Zhang, in the increasingly refined classification of gravitational instantons (cf. Collins-Jacob-Lin, Lee-Lin), and other areas. We show that these elliptic operators can be analyzed using the $\mathbf{a}$-pseudodifferential calculus of Grieser and Hunsicker, and use this to determine the space of $L^2$ harmonic forms in the four-dimensional setting, as well as the refined asymptotic regularity and local deformation theory of ALH* structures.