Minimal Energy Local Systems on Curves

Abstract

Let $\Sigma_{g,n}$ be an orientable topological surface of genus $g$ with $n$ punctures. When $g = 0$, Deroin and Tholozan studied the class of supra-maximal representations $\pi_1(\Sigma_{0,n})\to \mathrm{PSL}_2(\mathbb{R})$, and they showed that the supra-maximal representations form a compact component of a real relative character variety. We study a collection of rank $r$ local systems on $\Sigma_{g,n}$ which we call of minimal energy. These are generalizations of supra-maximal representations, and underlie polarizable complex variations of Hodge structure for any choice of complex structure on $\Sigma_{g,n}$. Like the supra-maximal representations, the minimal energy local systems form compact components of relative character varieties of real forms of $\mathrm{GL}_r(\mathbb{C})$. We show that when the local monodromy data around the punctures is chosen to be unitary and generic, and the relative character variety is nonempty, these minimal energy local systems always exist. When $g > 0$, we show that the minimal energy local systems come from unitary representations of $\pi_1(\Sigma_{g,n})$. If $g = 0$ we show that they do not always come from unitary representations, and we study their structure in general.