High-energy harmonic maps and degeneration of minimal surfaces

Let S be a closed surface of genus g > 2 and p a maximal PSL(2, R) x PSL(2, R) surface group representation. By a result of Schoen, there is a unique p-equivariant minimal surface & DAG;z in H2 x H2. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. We prove a similar result for maximal surfaces in AdS3. In the second half of the paper, we provide a geometric interpretation: the minimal surfaces & DAG;z degenerate to the core of a product of two R-trees. As a consequence, we obtain a compactification of the space of maximal representations of n -1(S) into PSL(2, R) x PSL(2, R).