Length spectrum compactification of the SO0(2,3)-Hitchin component

Abstract

We find a compactification of the SO0(2,3)-Hitchin component by studying the degeneration of the induced metric on the unique equivariant maximal surface in the 4-dimensional pseudo-hyperbolic space H2,2. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic quartic differentials on a Riemann surface. As an application, we describe the behavior of the entropy of the induced metric along rays of quartic differentials.