Metric Degenerations and Satake Compactification of Enriques Surfaces

Abstract

We determine the Gromov--Hausdorff compactification of the moduli space of unit-diameter Ricci-flat K\"ahler metrics on Enriques surfaces. We show that all metric degenerations are encoded by the adjoint Satake compactification of the Enriques period domain. There are 32 boundary components added in this Satake compactification, and the GH limit at each boundary is determined. As a corollary, all the GH limits of K\"ahler Ricci-flat metrics on Enriques surfaces are classified, including those for a fixed complex structure or a fixed polarization.