High-energy harmonic maps and degeneration of minimal surfaces

Let S be a closed surface of genus (formula presented) and ρ a maximal (formula presented) surface group representation. By a result of Schoen, there is a unique ρ –equivariant minimal surface (formula presented) 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 (formula presented) In the second half of the paper, we provide a geometric interpretation: the minimal surfaces (formula presented) 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 (formula presented).