Asymptotic behaviour of large-scale solutions of Hitchin's equations in higher rank

Takuro Mochizuki; Szilárd Szabó

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arXiv · 2303.04913

Asymptotic behaviour of large-scale solutions of Hitchin's equations in higher rank

Takuro Mochizuki·Szilárd Szabó

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Abstract

Let $X$ be a compact Riemann surface. Let $(E,\theta)$ be a stable Higgs bundle of degree $0$ on $X$. Let $h_{\det(E)}$ denote a flat metric of the determinant bundle $\det(E)$. For any $t>0$, there exists a unique harmonic metric $h_t$ of $(E,\theta)$ such that $\det(h_t)=h_{\det(E)}$. We prove that if the Higgs bundle is induced by a line bundle on the normalization of the spectral curve, then the sequence $h_t$ is convergent to the naturally defined decoupled harmonic metric at the speed of the exponential order. We also obtain a uniform convergence for such a family of Higgs bundles.

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BibTeX RIS

Takuro Mochizuki, Szilárd Szabó. 2023-03-08. Asymptotic behaviour of large-scale solutions of Hitchin's equations in higher rank. https://arxiv.org/abs/2303.04913

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