Proper superminimal surfaces of given conformal types in the hyperbolic four-space

TL;DR

This paper proves that any conformal superminimal immersion of a bordered Riemann surface into hyperbolic four-space can be approximated by proper immersions, using holomorphic Legendrian curves in the twistor space.

Contribution

It establishes the existence and approximation of proper conformal superminimal surfaces in hyperbolic four-space for any finite topological type Riemann surface.

Findings

Any bordered Riemann surface can be approximated by proper superminimal immersions in H^4.

Proper superminimal surfaces exist for any finite topological type Riemann surface.

The proof employs analysis of Legendrian curves in the twistor space.

Abstract

Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline M\to H^4$ can be approximated uniformly on compacts in $M$ by proper conformal superminimal immersions $M\to H^4$. In particular, $H^4$ contains properly immersed conformal superminimal surfaces normalised by any given open Riemann surface of finite topological type without punctures. The proof uses the analysis of holomorphic Legendrian curves in the twistor space of $H^4$.