A Calabi-Yau Metric on the Kummer Surface

Abstract

We prove the existence of a Ricci flat metric on the Kummer K3 surface. The proof follows the general strategy of Donaldson's gluing construction. However, we tackle the analysis without appealing to weighted norms or conformal transformations to model spaces, instead relying solely on compact elliptic theory on usual H\"older and Sobolev spaces. As the Eguchi-Hanson space plays a central role in the construction, we also present and compare different descriptions of this space, showing explicitly that it is isometric to a suitable Gibbons-Hawking ansatz.