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
