[Paper Review] Embedded minimal disks

Tobias Colding, William P. Minicozzi|ArXiv.org|Jun 14, 2002

Geometric Analysis and Curvature Flows13 references16 citations

TL;DR

This paper establishes that embedded minimal disks in R³ either behave like minimal graphs or resemble pieces of the helicoid, depending on curvature behavior. The key result shows that when curvature blows up along a Lipschitz curve (the singular set), the disks consist of two spiraling multi-valued graphs converging to a foliation of parallel planes away from the curve, with curvature divergence along the curve and non-removable singularities in the limit.

ABSTRACT

This is a survey of our work on embedded minimal disks.

Motivation & Objective

• To classify the asymptotic behavior of embedded minimal disks in R³ under blow-up of curvature.
• To understand the structure of singular limits of sequences of embedded minimal disks with unbounded curvature.
• To determine whether singularities in the limit lamination of minimal disks are removable or not.
• To construct explicit examples of embedded minimal disks with non-removable singularities using the Weierstrass representation.
• To establish the existence and regularity of multi-valued minimal graphs in the context of minimal surface theory.

Proposed method

• Use of multi-valued graphs to model helicoid-like behavior in embedded minimal disks, where each sheet is a single-valued graph over a sector of a punctured disk.
• Application of the one-sided curvature estimate to control the geometry of minimal surfaces near singularities.
• Employment of the Weierstrass representation to explicitly construct minimal immersions with desired curvature and embedding properties.
• Construction of a one-parameter family of minimal immersions using holomorphic data: $ g = e^{ih_a} $, $ \phi = dz $, and a simply connected domain $ \Omega_a $.
• Proof of embedding by showing vertical line segments in the domain map to curves that are graphs over a fixed line segment in a horizontal plane.
• Use of the intrinsic geometry and curvature estimates to analyze convergence to a lamination and to distinguish between removable and non-removable singularities.

Experimental results

Research questions

• RQ1 Under what conditions does a sequence of embedded minimal disks with unbounded curvature converge to a foliation of parallel planes?
• RQ2 Can the singular set in the limit of such sequences be a Lipschitz curve, and what geometric structure does it induce?
• RQ3 What determines whether a singularity in the limit lamination of embedded minimal disks is removable or not?
• RQ4 How can explicit minimal immersions be constructed to realize non-removable singularities in the limit?
• RQ5 What role do multi-valued graphs play in modeling the helicoid and its rescalings in the context of minimal surface limits?

Key findings

• Every embedded minimal disk with curvature blowing up along a curve is composed of exactly two multi-valued graphs spiraling into a Lipschitz singular curve.
• The limit of such sequences, away from the singular curve, is a foliation by parallel planes in $ \mathbb{R}^3 $, converging in $ C^\alpha $ topology.
• Curvature blows up along the entire singular curve: $ \sup_{B_r(\mathcal{S}(t)) \cap \Sigma_j} |A|^2 \to \infty $ for all $ r > 0 $, $ t \in \mathbb{R} $.
• A non-removable singularity occurs at the origin in the local example: the limit lamination on $ B_1 \setminus \{0\} $ does not extend to a lamination on $ B_1 $.
• The constructed sequence of minimal disks $ \Sigma_i \subset B_1 $ has curvature blowing up only at the origin, with $ \lim_{i \to \infty} |A|^2(0) = \infty $.
• The limit lamination consists of two spiraling minimal disks $ \Sigma^\pm $ and the punctured plane $ \{x_3 = 0\} \setminus \{0\} $, with $ \Sigma_i \setminus \{x_3 = 0\} $ converging to $ \Sigma^+ \cup \Sigma^- $.