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arXiv · 2010.05401

Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces

Qiongling Li·Takuro Mochizuki
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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.

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Qiongling Li, Takuro Mochizuki. 2020-10-12. Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces. https://arxiv.org/abs/2010.05401

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