[Paper Review] Null holomorphic curves in $\mathbb C^3$ and the conformal Calabi-Yau problem

Antonio Alarcón, Franc Forstnerič|arXiv (Cornell University)|Nov 8, 2013

Holomorphic and Operator Theory19 references3 citations

TL;DR

This paper establishes the existence of complete, properly immersed null holomorphic curves in ℂ³ with arbitrary topology, using a Runge-Mergelyan-type approximation theorem for null curves and a novel gluing method for sprays of holomorphic maps. The key result is the construction of complete bounded null curves in ℂ³, resolving a long-standing question in the conformal Calabi-Yau problem and extending to minimal surfaces in ℝ³ and Bryant surfaces in ℍ³.

ABSTRACT

In this paper we survey some recent contributions by the authors to the theory of null holomorphic curves in the complex Euclidean space $\mathbb C^3$, as well as their applications to null holomorphic curves in the special linear group $SL_2(\mathbb C)$, minimal surfaces in the Euclidean space $\mathbb R^3$, and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic 3-space $\mathbb H^3$. The paper is an expanded version of the lecture given by the second named author at the Abel Symposium in Trondheim, Norway, in July 2013.

Motivation & Objective

• To resolve the conformal Calabi-Yau problem for null holomorphic curves in ℂ³ by constructing complete, properly immersed null curves with arbitrary topology.
• To extend the theory of null curves to include bounded and complete immersions in ℂ³, addressing a long-standing open problem in minimal surface theory.
• To establish a Runge-Mergelyan-type approximation theorem for null curves in ℂ³, enabling the construction of null curves with prescribed geometric and topological properties.
• To apply the results to minimal surfaces in ℝ³ and constant mean curvature one surfaces (Bryant surfaces) in ℍ³, linking complex analysis to differential geometry.
• To prove that every open Riemann surface with non-constant bounded harmonic functions admits a complete conformal minimal immersion into ℝ³, generalizing earlier results.

Proposed method

• Utilizes a Runge-Mergelyan-type theorem for null curves in ℂ³ to approximate holomorphic maps into the null quadric 𝔸* = {z ∈ ℂ³ : z₁² + z₂² + z₃² = 0, z ≠ 0}.
• Employs holomorphic sprays of maps fₜ: M → 𝔸* parameterized by t ∈ B ⊂ ℂ^N, with the period map 𝒫(fₜ) submersive at t = 0 to ensure vanishing periods.
• Applies a Riemann-Hilbert boundary value problem technique to deform null discs near the boundary while preserving nullity and controlling the C⁰ norm of the third component.
• Uses an alternating deformation process on small boundary-adjacent discs, successively deforming in orthogonal null directions (e.g., V₁ = (1, i, 0), V₂ = (1, -i, 0)) with minimal perturbation to the third component.
• Employs a gluing method for sprays to patch local null curve deformations on a bordered Riemann surface M, ensuring global holomorphicity and vanishing periods.
• Applies the method to construct a global null curve G: M̄ → ℂ³ via integration G(z) = F(p) + ∫ₚᶻ gθ, where g: M̄ → 𝔸* has vanishing periods and integrates to a null curve.

Experimental results

Research questions

• RQ1 Do there exist complete, properly immersed null holomorphic curves in ℂ³ with arbitrary topology and bounded image?
• RQ2 Can the Calabi-Yau problem for minimal surfaces in ℝ³ be extended to the complex setting via null curves in ℂ³?
• RQ3 Is it possible to construct complete bounded null curves in ℂ³ using approximation techniques that preserve the null condition?
• RQ4 Can the period problem for null curves be solved via a submersive period map and deformation of sprays on bordered Riemann surfaces?
• RQ5 What is the relationship between the existence of bounded harmonic functions on a Riemann surface and the existence of complete bounded minimal immersions into ℝ³?

Key findings

• The paper constructs, for any open bordered Riemann surface M, a complete, properly immersed null holomorphic curve F: M̄ → ℂ³ with image contained in any given convex domain of ℂ³.
• It proves the existence of complete bounded null curves in ℂ³, answering a question posed by Martín, Umehara, and Yamada and resolving a key case of the conformal Calabi-Yau problem.
• The construction relies on a Runge-Mergelyan-type theorem for null curves, which allows approximation of holomorphic maps into the null quadric 𝔸* with vanishing periods.
• The method enables the construction of complete minimal surfaces in ℝ³ with bounded image, extending Nadirashvili’s result to arbitrary topology.
• It is shown that every open Riemann surface admitting non-constant bounded harmonic functions supports a complete conformal minimal immersion into ℝ³.
• The alternating deformation process in orthogonal null directions, with controlled C⁰ perturbation of the third component, leads to convergence to a proper null curve with bounded third coordinate.