UC Berkeley

Hyperkähler manifolds are one of the simplest examples of Einstein manifolds. They are Ricci-flat Riemannian manifolds with special holonomy. In dimension 4, hyperkähler 4-manifolds can be purely described by a triple of symplectic 2-forms that satisfy the pointwise orthonormal condition with respect to the wedge product.

In this dissertation, we proved the compactness of a set of hyperkähler 4-manifolds with boundary under Cheeger-Gromov topology, where we assume only geometric control on the boundary and topological conditions. We showed that our proof can be extended to Einstein 4-manifolds with boundary by assuming only additional topological conditions.

Furthermore, we discuss the period map for K3 surfaces in a differential geometric setting. We gave a simple proof for the surjectivity of the period map, without invoking Yau's theorem on the Calabi conjecture or any algebraic geometry. The key is to show that when a sequence of hyperkähler metrics has bounded period in some sense, then the sequence has a convergent subsequence under Cheeger-Gromov topology.