[Paper Review] Willmore flow of surfaces in Riemannian spaces I: Concentration-compactness

Jan Metzger, Glen Wheeler+1 more|arXiv (Cornell University)|Aug 28, 2013

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper establishes concentration-compactness principles for the Willmore flow of closed surfaces in Riemannian 3-manifolds, proving lower bounds on the maximal existence time of the flow based on initial concentration of curvature and area. It provides two lifespan theorems: one for non-positively curved ambient spaces using only curvature concentration, and another for general spaces requiring control of both curvature and area concentration, ensuring long-time existence under small initial data conditions.

ABSTRACT

In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal time of existence of the flow proportional to the concentration of the curvature and area at initial time. The estimate from the first theorem is purely in terms of the concentration of curvature at initial time but applies only to ambient spaces with non-positive sectional curvature. The second requires additional information on the concentration of area at initial time but applies in more general background spaces. Applications of these results shall appear in forthcoming work.

Motivation & Objective

• To establish lifespan theorems for the Willmore flow of closed surfaces in Riemannian 3-manifolds using concentration-compactness principles.
• To derive lower bounds on the maximal existence time of the Willmore flow based on initial concentration of curvature and area.
• To extend existence results to general Riemannian ambient spaces by incorporating area concentration when curvature control alone is insufficient.
• To provide quantitative estimates on curvature and area concentration that guarantee long-time smooth existence of the flow.
• To generalize prior results on Willmore flow in Euclidean space to curved Riemannian ambient spaces with controlled curvature and injectivity radius.

Proposed method

• Formulate the Willmore flow as the $L^2$-gradient flow of the $L^2$-norm of mean curvature, leading to a fourth-order quasilinear parabolic PDE for the immersion.
• Use local existence theory for higher-order degenerate quasilinear parabolic equations to establish short-time smooth existence of the flow.
• Apply covering arguments and Sobolev inequalities in the ambient manifold to control curvature concentration, relying on bounds on Ricci curvature and its derivatives up to order 5.
• Introduce a smallness condition on the $L^2$-norm of the second fundamental form $|A|^2$ over balls of radius $ ho$ in the ambient space to control concentration.
• Use a continuity argument with a maximal time $t_0$ where curvature concentration remains bounded, and derive a contradiction if $t_0 < ext{expected lifespan}$, proving the lifespan estimate.
• In the general case, supplement curvature concentration with area concentration via a new control estimate (Proposition 17) to compensate for failure of Sobolev inequalities in positive curvature regions.

Experimental results

Research questions

• RQ1 What is the maximal time of existence for the Willmore flow of a closed surface in a Riemannian 3-manifold, given initial concentration of curvature?
• RQ2 How does the ambient manifold's curvature (especially non-positive sectional curvature) affect the lifespan of the Willmore flow?
• RQ3 Can lifespan estimates be extended to general Riemannian 3-manifolds with positive curvature by incorporating area concentration?
• RQ4 Under what conditions does the Willmore flow remain smooth for a time proportional to $ ho^4$, where $ ho$ is the radius of curvature concentration?
• RQ5 What role does the injectivity radius and higher-order derivatives of Ricci curvature play in the long-time existence of the Willmore flow?

Key findings

• For Riemannian 3-manifolds with non-positive sectional curvature, the maximal existence time $T$ of the Willmore flow satisfies $T /geq rac{1}{c} ho^4$, where $c$ depends only on the ambient metric and bounds on $ abla^{(k)} ext{Ric}$ for $k=0$ to $5$.
• The curvature concentration $ ho$-ball $L^2$-norm of $|A|^2$ remains bounded by $c imes ext{initial concentration}$ for all $t ext{ in } [0, rac{1}{c} ho^4]$, ensuring regularity.
• In general Riemannian 3-manifolds with positive injectivity radius, lifespan is bounded below by $T ext{ in } [0, rac{1}{c} ho^4]$ provided both curvature and area concentration are initially small.
• The lifespan estimate is robust under small perturbations of initial data, as long as the initial $|A|^2$ and area concentration in $ ho$-balls are sufficiently small.
• The proof relies on a contradiction argument using continuity of the curvature concentration functional and compactness theorems, showing that finite-time blowup cannot occur under the smallness assumptions.
• The constants in the lifespan estimate depend only on the ambient metric and the $L^rown{ ext{infty}}$-norms of Ricci curvature and its derivatives up to order 5, not on the initial surface geometry beyond concentration.