Diagrams and harmonic maps, revisited

TL;DR

This paper generalizes harmonic map results from the 2-sphere to arbitrary Riemann surfaces, introducing a new nilpotent cycle theory and applying it to classify minimal surfaces and harmonic maps of finite type.

Contribution

It develops a novel nilpotent cycle framework for harmonic maps of finite uniton number from any Riemann surface, extending previous results.

Findings

Extended harmonic map results to arbitrary Riemann surfaces

Developed a new nilpotent cycle theory for these maps

Provided classification of minimal surfaces of constant curvature

Abstract

We extend many known results for harmonic maps from the 2-sphere into a Grassmannian to harmonic maps of finite uniton number from an arbitrary Riemann surface. Our method relies on a new theory of nilpotent cycles arising from the diagrams of F.E.~Burstall and the second author associated to such harmonic maps; these properties arise from a criterion for finiteness of the uniton number found recently by the authors with A.~Aleman. Applications include a new classification result on minimal surfaces of constant curvature and a constancy result for finite type harmonic maps.