[Paper Review] Geometry of Two-dimensional Self-shrinkers

Lu Wang|arXiv (Cornell University)|May 1, 2015

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper establishes a local graphical theorem for 2-dimensional self-shrinkers away from the origin, proving that under small mean curvature and bounded second fundamental form, the surface is locally a graph with small gradient. As applications, it proves uniform boundedness of the second fundamental form for noncompact self-shrinkers with bounded mean curvature and locally finite genus, and confirms asymptotic cylindrical or conical behavior at infinity for finite genus noncompact self-shrinkers.

ABSTRACT

We prove a local graphical theorem for two-dimensional self-shrinkers away from the origin. As applications, we study the asymptotic behavior of noncompact self-shrinkers with finite genus. Also, we show uniform boundedness on the second fundamental form of two-dimensional noncompact self-shrinkers with bounded mean curvature and uniform locally finite genus.

Motivation & Objective

• To establish a local graphical decomposition theorem for 2-dimensional self-shrinkers away from the origin, enabling analysis of asymptotic geometry.
• To address Conjecture 1.1 on the asymptotic structure of noncompact self-shrinkers with finite genus, particularly their convergence to cones or cylinders at infinity.
• To prove uniform boundedness of the second fundamental form for noncompact 2D self-shrinkers with bounded mean curvature and locally finite genus.
• To confirm the smooth convergence of translated self-shrinkers to a plane or cylinder, as required by Conjecture 1.2.
• To use maximum principle arguments and geometric analysis to rule out Catenoid-like limiting shapes in the asymptotic regime.

Proposed method

• Establish a local graphical theorem (Theorem 1.3) under conditions of small mean curvature and bounded $L^2$-norm of the second fundamental form.
• Use a bootstrap argument and exploit the dual scaling behavior of the radial direction as both a spatial and time-like direction in the mean curvature flow.
• Apply the compactness theorem of Brakke and White's stratification theorem to extract subsequences converging to minimal surfaces with finite total curvature.
• Classify the limit minimal surface as a Catenoid via the classification theorem of minimal surfaces with finite total curvature and genus zero.
• Use maximum principles on $|\mathbf{x}|^2$ to rule out the existence of Catenoid-like limits, deriving a contradiction via connected component counting.
• Employ geometric comparison via normal graphs and transversality arguments to show that multiple distinct cones or cylinders cannot coexist at infinity.

Experimental results

Research questions

• RQ1 Under what conditions is a 2-dimensional self-shrinker locally graphical away from the origin?
• RQ2 Can noncompact self-shrinkers with finite genus be shown to asymptotically approach a cone or a cylinder at infinity?
• RQ3 Is the second fundamental form uniformly bounded for noncompact 2D self-shrinkers with bounded mean curvature and locally finite genus?
• RQ4 Does the normalized mean curvature flow of a self-shrinker converge smoothly to a plane or cylinder under suitable conditions?
• RQ5 Can the Catenoid be ruled out as a limit of the rescaled self-shrinker at infinity using maximum principle arguments?

Key findings

• A local graphical theorem is proven: under small mean curvature and bounded $L^2$-norm of the second fundamental form, the surface is locally a graph with gradient bounded by $\delta_0$.
• For noncompact properly embedded self-shrinkers with finite genus, the ends are asymptotically cylindrical or conical, confirming part of Conjecture 1.1.
• The second fundamental form is uniformly bounded for noncompact 2D self-shrinkers with bounded mean curvature and locally finite genus.
• The limit of $\Sigma - \lambda \mathbf{v}_0$ as $\lambda \to \infty$ converges locally smoothly to a plane or self-shrinking cylinder of multiplicity one.
• A Catenoid cannot arise as a limit of rescaled self-shrinkers due to a contradiction in the number of connected components when applying the maximum principle to $|\mathbf{x}|^2$.
• The proof relies on contradiction via connected component counting after constructing multiple transverse cones or cylinders, showing that such configurations are incompatible with finite genus and proper embedding.