Members & Former Members

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\mathrm{dS}_3$, best viewed as a generalized Gauß map. On the gauge theoretic side, it matches $\mathrm{SU}(2)$ and $\mathrm{SU}(1,1)$ solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of \emph{transgressive} harmonic maps $f:M\to \mathbb S^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $\tau$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines.

Related project(s):
55New hyperkähler spaces from the the self-duality equations77Asymptotic geometry of the Higgs bundle moduli space II