  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.
