# Embedded $H$-Planes in Hyperbolic 3-Space

## TL;DR

This paper proves the existence and uniqueness of embedded constant mean curvature (H) surfaces in hyperbolic 3-space with prescribed asymptotic boundaries, extending the understanding of geometric structures in hyperbolic geometry.

## Contribution

It establishes the existence of embedded H-planes with arbitrary C^0 Jordan curves at infinity and proves uniqueness for generic boundary curves, advancing the theory of H-surfaces in hyperbolic space.

## Key findings

- Existence of embedded H-planes for any C^0 Jordan curve at infinity.
- Existence of a unique minimizing H-plane for generic boundary curves.
- Any quasi-Fuchsian manifold contains an H-surface in the homotopy class of the core surface.

## Abstract

We show that for any C^0 Jordan curve C in the sphere at infinity of H^3, there exists an embedded $H$-plane P_H in H^3 with asymptotic boundary C for any H in (-1,1). As a corollary, we proved that any quasi-Fuchsian hyperbolic 3-manifold M=SxR contains an H-surface S_H in the homotopy class of the core surface S for any H in (-1,1). We also proved that for any C^1 Jordan curve J in the sphere at infinity, there exists a unique minimizing H-plane P_H with asymptotic boundary J for a generic H in (-1,1).