Equivariant isometric immersions of surfaces in hyperbolic space
Abstract
Let S be a closed, oriented surface of genus at least $2$, let h be a smooth Riemannian metric on S with curvature K∈ (−1,0], and let c∈ cTS be a conformal structure on S. There exists a unique equivariant isometric immersion of (S,h) in HH3 such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is c. Dually, if h* is a smooth metric on S with curvature K*∈ (−∞, 0) and if c∈ cTS, there exists a unique equivariant immersion of S into HH3 with third fundamental form h* and such that the pull-back of the conformal class at infinity by the Gauss map is c. Equivalently, given h and c∈ cTS, there is a unique pair (E,u) where E is a hyperbolic end with conformal structure at infinity c and u is an isometric embedding of (S,h) in E. Given h* and c, there exists a unique pair (E,u*), where E is a hyperbolic end with conformal structure at infinity c, and u*:S→ E is an embedding inducing the third fundamental form h*. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.
Related
- Non-maximal quasi-isometric PU2,n+1-representations via alternating surfaces in complex pseudo-hyperbolic spaces
- Isometric Embeddings of Asymptotically Hyperbolic Manifolds into Hyperbolic Space
- ℝ-equivariant minimal Lagrangian surfaces in the complex quadric Q2 and the complex hyperbolic quadric Q2*
- Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature
- Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space
BibTeX
@misc{schlenker2026equivariant,
title = {{Equivariant isometric immersions of surfaces in hyperbolic space}},
author = {Jean-Marc Schlenker},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32300},
eprint = {2609.32300},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32300},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equivariant isometric immersions of surfaces in hyperbolic space
AU - Jean-Marc Schlenker
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32300
UR - https://arxiv.org/abs/2609.32300
AB - Let S be a closed, oriented surface of genus at least $2$, let h be a smooth Riemannian metric on S with curvature K∈ (−1,0], and let c∈ cT_S be a conformal structure on S. There exists a unique equivariant isometric immersion of (S,h) in HH^3 such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is c. Dually, if h^* is a smooth metric on S with curvature K^*∈ (−∞, 0) and if c∈ cT_S, there exists a unique equivariant immersion of S into HH^3 with third fundamental form h^* and such that the pull-back of the conformal class at infinity by the Gauss map is c. Equivalently, given h and c∈ cT_S, there is a unique pair (E,u) where E is a hyperbolic end with conformal structure at infinity c and u is an isometric embedding of (S,h) in E. Given h^* and c, there exists a unique pair (E,u^*), where E is a hyperbolic end with conformal structure at infinity c, and u^*:S→ E is an embedding inducing the third fundamental form h^*. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.
ER -