Singularity Analysis of Curvature Flow of Curves on a Riemannian Surface

Abstract

In this paper, we consider evolution of embedded curves by curvature flow in a compact Riemannian surface. Let 𝛾 be a closed embedded curve evolving under the curvature flow in a compact surface M. If a singularity develops in finite time, then the curve shrinks to a point. Therefore, when t is close enough to the blow- up time, we may assume that the curve is contained in a small neighborhood of the collapsing point on the surface. Using a local conformal diffeomorphism 𝜙:𝑈 ⊆ 𝑀 → 𝑈 ′ ⊆ ℝ 2 between compact neighborhoods, we get a corresponding flow in the plane which satisfies the following equation: 𝜕𝛾 ′ 𝜕𝑡 = ( 𝑘 ′ 𝐽 2 − ∇𝑁 𝐽 𝐽 2 )𝑁 ′ where 𝛾 ′ 𝑝,𝑡 = 𝜙 𝛾 𝑝,𝑡 , 𝑘 ′ is the curvature of 𝛾 ′ in 𝑈 ′ , 𝑁 ′ is the unit normal vector, and the conformal factor J is smooth, bounded and bounded away from 0. We define the extrinsic and intrinsic distance functions 𝑑, 𝑙 ∶ Γ × Γ × 0, 𝑇 → ℝ by 𝑑 𝑝, 𝑞,𝑡 ≔ 𝛾 𝑝,𝑡 − 𝛾 𝑞,𝑡 ℝ 2 and 𝑙 𝑝, 𝑞,𝑡 ≔ 𝑑𝑠𝑡 = 𝑠𝑡 𝑞 − 𝑠𝑡 𝑝 𝑞 𝑝 where Γ is either S 1 or an interval. We also define the smooth function ψ: S 1 × S 1 × [0, T] → ℝ by 𝜓 𝑝, 𝑞,𝑡 ≔ 𝐿(𝑡) 𝜋 sin 𝑙 𝑝, 𝑞,𝑡 𝜋 𝐿 𝑡 . We use the distance comparison 𝑑 𝑙 and 𝑑 𝜓 to prove the following theorem. Main Theorem: Let 𝛾 be a closed embedded curve evolving by curvature flow on a smooth compact Riemannian surface. If a singularity develops in finite time, then the curve converges to a round point in the 𝐶 ∞ sense. This extends Huisken's distance comparison technique for curvature flow of embedded curves in the plane. Hamilton used isoperimetic estimates techniques to prove that when a closed embedded curve in the plane evolves by curvature flow the curve converges to a round point and Zhu used Hamilton's isoperimetric estimates techniques to study asymptotic behavior of anisotropic curves flows