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arXiv · 2508.10335

Harmonic maps and framed $\mathrm{PSL}_2(\mathbb{C})$-representations

Subhojoy Gupta·Gobinda Sau
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Abstract

We show that given an element $X$ of the enhanced Teichm\"{u}ller space $\mathcal{T}^\pm(\mathbb{S}, \mathbb{M})$ and a type-preserving framed $\mathrm{PSL}_2(\mathbb{C})$-representation $\hat{\rho} = (\rho,\beta)$, there is a $\rho$-equivariant harmonic map $f:\mathbb{H}^2 \to \mathbb{H}^3$ that is asymptotic to the framing $\beta$. Here, the domain is the universal cover of the punctured Riemann surface obtained from a conformal completion of $X$. Moreover, such a harmonic map is unique if one prescribes, in addition, the principal part of the Hopf differential at each puncture. The proof uses the harmonic map heat flow.

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Subhojoy Gupta, Gobinda Sau. 2025-08-14. Harmonic maps and framed $\mathrm{PSL}_2(\mathbb{C})$-representations. https://arxiv.org/abs/2508.10335

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