Geometry & Topology Seminar: Equivariant conformal harmonic maps from surfaces into rank 2 real buildings

Abstract:

In joint work with Andrea Tamburelli and Mike Wolf, we produce many equivariant minimal embeddings f of the universal cover of a closed oriented surface S of genus at least 2 into the asymptotic cone B of the symmetric space X=SL(3,R)/SO(3).  More specifically, we use a ray of cubic differentials {tQ}, related Higgs bundles and the nonlinear Hodge correspondence to produce a representation of  π₁S into SL(3,R) and an equivariant conformal harmonic map into X. Then take the limit t → ∞.  The geometry of the image of  f can be read off explicitly from a flat structure induced by the cubic differential.

 

More generally, we can consider B to be a real building of rank 2.  We address the question of uniqueness of equivariant conformal harmonic maps into a real building B of rank 2.  In the case of SL(3,R) and an action of the fundamental group of a surface into the isometries of B, we can show uniqueness for generic equivariant conformal harmonic maps.  We discuss in some detail the geometry of the spaces involved and make a conjecture about the general case.  This is joint work in progress with Tamburelli and Wolf.

Host: Charles Ouyang