Work

Complete Solutions of Toda Equations and Cyclic Higgs Bundles Over Non-compact Surfaces

Year: 2025

Type: article

Abstract: On a Riemann surface with a holomorphic $r$-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a harmonic metric of the Higgs bundle for which the grading is orthogonal. Here we focus on a general non-compact Riemann surface with an $r$-differential which is not necessarily meromorphic at infinity. We introduce the notion of complete solution of the Toda equation, and we prove the existence and uniqueness of a complete solution by using techniques for both Toda equations and harmonic bundles. Moreover, we show some quantitative estimates of the complete solution.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf081

Authors: Qiongling Li, Takuro Mochizuki

Institutions: Nankai University, Kyoto University, Research Institute for Mathematical Sciences, Kyoto University

Language: English

Topic: Geometry and complex manifolds

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Award: Algebraic Geometry and Integrable Systems -- Moduli theory and Equations of Painleve type