Harmonic metrics of generically regular semisimple Higgs bundles on non-compact Riemann surfaces

TL;DR

This paper proves the existence and uniqueness of harmonic metrics compatible with symmetric pairings on generically regular semisimple Higgs bundles over non-compact Riemann surfaces, including classification results.

Contribution

It establishes the existence of harmonic metrics compatible with symmetric pairings on such Higgs bundles and classifies these metrics on punctured Riemann surfaces, proving uniqueness in the wild case.

Findings

Existence of compatible harmonic metrics for generically regular semisimple Higgs bundles.

Uniqueness of harmonic metrics in the wild, regular semisimple case.

Classification results for harmonic metrics on punctured Riemann surfaces.

Abstract

We prove that a generically regular semisimple Higgs bundle equipped with a non-degenerate symmetric pairing on any Riemann surface always has a harmonic metric compatible with the pairing. We also study the classification of such compatible harmonic metrics in the case where the Riemann surface is the complement of a finite set $D$ in a compact Riemann surface. In particular, we prove the uniqueness of a compatible harmonic metric if the Higgs bundle is wild and regular semisimple at each point of $D$.