[Paper Review] Minimal cones and self-expanding solutions for mean curvature flows

Qi Ding|arXiv (Cornell University)|Mar 9, 2015

Geometric Analysis and Curvature Flows31 references3 citations

TL;DR

This paper establishes the existence and uniqueness of smooth, embedded self-expanding hypersurfaces in the exterior of a $C^{3,ar{\alpha}}$ mean convex but not area-minimizing cone in $\mathbb{R}^{n+1}$, using weighted $L^2$ minimization and barrier techniques. The key result confirms Lawson's conjecture for regular minimal but non-area-minimizing cones by constructing a foliation of the domain via such self-expanders with positive mean curvature.

ABSTRACT

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $C^{3,α}$-regular and mean convex (but not area-minimizing), we can prove that the corresponding self-expanding hypersurfaces are smooth, embedded, and have positive mean curvature everywhere (see Theorem 1.1). As a result, for regular minimal but not area-minimizing cones we can give an affirmative answer to a problem arisen by Lawson [4].

Motivation & Objective

• To resolve a problem posed by Lawson regarding the existence of smooth, embedded hypersurfaces with positive mean curvature in a neighborhood of a minimal cone.
• To establish the existence of $E$-minimizing self-expanding hypersurfaces with prescribed $C^{3,\alpha}$ mean convex, non-area-minimizing tangent cones at infinity.
• To prove that such self-expanders are smooth, embedded, and form a foliation of the domain via rescaling.
• To show that uniqueness fails for area-minimizing cones, but holds for non-minimizing ones under the given regularity and convexity conditions.

Proposed method

• Constructing rotational symmetric graphic self-expanders as barriers to control the asymptotic behavior of solutions.
• Using weighted $L^2$ minimization of the functional $\int_K e^{|X|^2/4} d\mu$ to define $E$-minimizing currents.
• Applying Allard's regularity theorem to establish $C^{3,\alpha}$ regularity of the limit current outside a large ball.
• Employing maximum principle arguments on the mean curvature equation $\frac{1}{2}\langle X, \nu \rangle = H$ to prove positivity and non-degeneracy of mean curvature.
• Using barrier comparison and projection techniques to rule out multiple solutions via contradiction on the intersection behavior of rescaled hypersurfaces.
• Analyzing asymptotic decay estimates of Jacobi fields at infinity to control the behavior of the self-expanding current near the cone.

Experimental results

Research questions

• RQ1 Can a smooth, embedded self-expanding hypersurface with positive mean curvature exist in the exterior of a $C^{3,\alpha}$ mean convex but not area-minimizing cone?
• RQ2 Does such a self-expander form a foliation of the domain under rescaling, i.e., $\sqrt{t}M$ for $t>0$?
• RQ3 Is the self-expander unique among all $E$-minimizing solutions with the same tangent cone at infinity?
• RQ4 What happens if the cone is area-minimizing? Does uniqueness still hold?
• RQ5 Can the singularities of $E$-minimizing currents be ruled out under mean convexity and non-minimality assumptions?

Key findings

• For any $C^{3,\alpha}$ mean convex but not area-minimizing cone $C$ in $\mathbb{R}^{n+1}$, there exists a unique smooth, complete, embedded $E$-minimizing self-expanding hypersurface $M$ in the domain $\Omega$ bounded by $C$.
• The self-expander $M$ has positive mean curvature everywhere and is properly embedded in $\Omega$.
• The rescaled family $\mathcal{M}: t \mapsto \sqrt{t}M$ forms a smooth foliation of $\Omega$ by mean curvature flow.
• The assumption that the cone is not area-minimizing is essential: if $C$ is area-minimizing, then any smooth self-expander converging to $C$ must be $C$ itself.
• Uniqueness fails in general without the non-minimality condition, as shown by examples in Ilmanen and others.
• The proof relies on barrier constructions, maximum principle arguments on the mean curvature equation, and asymptotic decay estimates of Jacobi fields at infinity.