Geodesics on a K3 Surface near the Orbifold Limit

TL;DR

This paper investigates the geometry of Kummer K3 surfaces near the orbifold limit, improving metric estimates and exploring conditions for the existence of stable closed geodesics, highlighting the influence of symmetry and hyperk"ahler identities.

Contribution

It refines Calabi-Yau metric estimates on K3 surfaces near the orbifold limit and analyzes the existence and restrictions of stable closed geodesics.

Findings

Improved estimates for Calabi-Yau metrics on K3 surfaces near the orbifold limit.

Identified restrictions on stable closed geodesics due to metric properties.

Demonstrated existence of stable geodesics in symmetric cases via hyperk"ahler identities.

Abstract

This article studies Kummer K3 surfaces close to the orbifold limit. We improve upon estimates for the Calabi-Yau metrics due to R. Kobayashi. As an application, we study stable closed geodesics. We use the metric estimates to show how there are generally restrictions on the existence of such geodesics. We also show how there can exist stable, closed geodesics in some highly symmetric circumstances due to hyperk\"{a}hler identities.