A Calabi-Yau Metric on the Kummer Surface

TL;DR

This paper establishes the existence of a Ricci-flat Calabi-Yau metric on the Kummer surface using a novel analytical approach that avoids weighted norms, relying instead on standard elliptic theory.

Contribution

It provides a new proof of the Ricci-flat metric existence on the Kummer surface, employing a purely elliptic theory approach without weighted norms or conformal transformations.

Findings

Constructed a Ricci-flat metric on the Kummer surface.

Demonstrated the Eguchi-Hanson space's isometry to a Gibbons-Hawking ansatz.

Compared different descriptions of the Eguchi-Hanson space.

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.