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Harmonic metrics of generically regular semisimple Higgs bundles on noncompact Riemann surfaces

Qiongling Li and Takuro Mochizuki

Vol. 5 (2023), No. 4, 663–711
Abstract

We prove that a generically regular semisimple Higgs bundle equipped with a nondegenerate 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.

Keywords
harmonic bundle, nondegenerate symmetric product, real structure
Mathematical Subject Classification
Primary: 53C07
Secondary: 58E15, 14D21, 81T13
Milestones
Received: 15 October 2022
Revised: 23 July 2023
Accepted: 9 August 2023
Published: 21 November 2023
Authors
Qiongling Li
Chern Institute of Mathematics and LPMC
Nankai University
Tianjin
China
Takuro Mochizuki
Research Institute for Mathematical Sciences
Kyoto University
Kyoto
Japan