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arXiv · 2610.35812

Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space

Jorge Hidalgo
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Abstract

We show that every open Riemann surface may be conformally properly embedded in the tangent sphere bundle of hyperbolic $3$-space as a Legendrian surface. We achieve this by constructing proper holomorphic Legendrian embeddings in $\mathrm{SL}_2 (\mathbb{C})$ whose projections to the quotient $\mathrm{SL}_2 (\mathbb{C})/ \mathrm{U}_1$ remain embeddings. As another consequence, every open Riemann surface admits a proper holomorphic Legendrian embedding in the projectivised special linear group $\mathrm{PSL}_2 (\mathbb{C})$.

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Jorge Hidalgo. 2026-10-26. Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space. https://arxiv.org/abs/2610.35812

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