The Calabi-Yau problem, null curves, and Bryant surfaces

TL;DR

This paper proves that any bordered Riemann surface can be properly embedded into complex 3-space with null holomorphic curves, leading to new examples of Bryant surfaces and minimal immersions with bounded images, using novel complex analytic methods.

Contribution

It introduces a new method based on complex analysis to embed bordered Riemann surfaces into complex space with null curves, maintaining a fixed conformal structure.

Findings

Existence of proper null holomorphic embeddings into ^3 with bounded images.

Construction of proper Bryant surfaces with finite topology.

Development of a new approach using Riemann-Hilbert boundary value problem solutions.

Abstract

In this paper we prove that every bordered Riemann surface M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean $3$-space $\mathbb{C}^3$. The real part of such an embedding is a complete conformal minimal immersion $M\to \mathbb{R}^3$ with bounded image. For any such $M$ we also construct proper null holomorphic embeddings $M\to \mathbb{C}^3$ with a bounded coordinate function; these give rise to properly embedded null curves $M\to SL_2(\mathbb{C})$ and to properly immersed Bryant surfaces $M\to \mathbb{H}^3$ in the hyperbolic $3$-space. In particular, we give the first examples of proper Bryant surfaces with finite topology and of hyperbolic conformal type. The main novelty when compared to the existing results in the literature is that we work with a fixed conformal structure on $M$. This is accomplished by introducing a conceptually new method based on complex analytic techniques. One of our main tools is an approximate solution to the Riemann-Hilbert boundary value problem for null curves in $\mathbb{C}^3$.