[Paper Review] Stability of spherical collapse under mean curvature flow

Israel Michael Sigal, Wenbin Kong|arXiv (Cornell University)|Oct 24, 2011

Geometric Analysis and Curvature Flows22 references3 citations

TL;DR

This paper establishes the asymptotic stability of spherical collapse under mean curvature flow in ℝⁿ⁺¹ for initial hypersurfaces sufficiently close to the standard n-sphere in the Hˢ norm with s > n/2 + 1. Using a nonlinear parabolic PDE framework and energy estimates in Sobolev spaces, it proves that such surfaces collapse to a round point in finite time, approaching spheres of radius √(2n(tₜ - t)) with exponentially fast convergence, confirming the stability of the spherical singularity scenario.

ABSTRACT

We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore it does not impose restrictions of the mean curvature of the initial surface. We show that the solution of such a flow collapses to a point, $z_*$, in a finite time, $t_*$, approaching exponentially fast the spheres of radii $\sqrt{2n(t_*-t)}$, centered at $z(t)$, with the latter converging to $z_*$. Keywords: mean curvature flow, evolution of surfaces, collapse of surfaces, asymptotic stability, asymptotic dynamics, dynamics of surfaces, mean curvature soliton, nonlinear parabolic equation.

Motivation & Objective

• To establish the stability of the spherical singularity under mean curvature flow in ℝⁿ⁺¹.
• To analyze the long-time dynamics of mean curvature flow for initial hypersurfaces near the standard n-sphere.
• To show that perturbations of the n-sphere evolve smoothly and collapse to a round point in finite time.
• To characterize the asymptotic approach to the singularity, including convergence rates of the center and radius.
• To remove restrictions on mean curvature by using high-order Sobolev norms (s > n/2 + 1).

Proposed method

• Formulates the mean curvature flow as a nonlinear parabolic PDE for the immersion X(ω,t) = z(t) + R(ω,t)ω.
• Imposes initial data in the Hˢ(Sⁿ) norm with s > n/2 + 1 to ensure regularity and avoid mean curvature constraints.
• Uses a decomposition of the evolving surface into radial deviation ξ(ω,t) and center z(t), with R(ω,t) = λ(t)(√(n/a(t)) + ξ(ω,t)).
• Applies energy estimates and Sobolev embeddings to control nonlinear terms in the evolution equation.
• Employs commutator estimates and fractional Sobolev norms to bound nonlinearities in the PDE.
• Establishes uniform bounds on the Hˢ norm of the deviation ξ(·,t) and controls the convergence of λ(t) and a(t) to their asymptotic values.

Experimental results

Research questions

• RQ1 Is the spherical collapse solution under mean curvature flow stable under small Hˢ perturbations with s > n/2 + 1?
• RQ2 Do initial hypersurfaces close to the n-sphere evolve smoothly and collapse to a point in finite time?
• RQ3 How fast do the evolving surfaces approach the shrinking spheres of radius √(2n(tₜ - t)) near the singularity?
• RQ4 What is the asymptotic behavior of the center z(t) and the radial profile R(ω,t) as t → tₜ?
• RQ5 Can the mean curvature of the initial surface be arbitrary, or does it impose constraints on the flow?

Key findings

• The solution collapses to a point z* in finite time t* < ∞, with t* → ∞ as the initial perturbation size → 0.
• The evolving surface Mₜ approaches spheres of radius √(2n(tₜ - t)) centered at z(t), with exponential convergence rate.
• The center z(t) converges to z* as O((tₜ - t)^(1/(2a*))^(n + 1/2 - 1/(2n))) with a* > 0.
• The radial profile R(ω,t) satisfies R(ω,t) = λ(t)(√(n/a(t)) + ξ(ω,t)), where λ(t) = √(2a*tₜ - t) + O((tₜ - t)^(1/2 + 1/(2a*))^(1 - 1/(2n)))
• The Hˢ norm of the deviation ξ(·,t) decays as ‖ξ(·,t)‖_{Hˢ} ≲ (tₜ - t)^(1/(2n)) as t → tₜ.
• The initial surface's mean curvature is not restricted, as the Hˢ norm with s > n/2 + 1 controls the flow without imposing curvature bounds.