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Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space
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 SL2 (ℂ) whose projections to the quotient SL2 (ℂ)/ U1 remain embeddings. As another consequence, every open Riemann surface admits a proper holomorphic Legendrian embedding in the projectivised special linear group PSL2 (ℂ).
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BibTeX
@misc{hidalgo2026properly,
title = {{Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space}},
author = {Jorge Hidalgo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35812},
eprint = {2610.35812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35812},
}
RIS
TY - UNPB
TI - Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space
AU - Jorge Hidalgo
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35812
UR - https://arxiv.org/abs/2610.35812
AB - 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 SL_2 (ℂ) whose projections to the quotient SL_2 (ℂ)/ U_1 remain embeddings. As another consequence, every open Riemann surface admits a proper holomorphic Legendrian embedding in the projectivised special linear group PSL_2 (ℂ).
ER -