Minimal surfaces in symmetric spaces

For S a closed surface of genus at least 2, Labourie proved that every Hitchin representation of pi_1(S) into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the space of Hitchin representations admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmuller space.

In this talk we will discuss the analysis and geometry of minimal surfaces in symmetric spaces, and explain how certain large area minimal surfaces give counterexamples to Labourie’s conjecture. This is all joint work with Peter Smillie.