Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem

By Ben Andrews, Paul Bryan. 2009

62 citations recorded by the source corpus.

Abstract A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length 2π. The estimate bounds the length of any chord from below in terms of the arc length between its endpoints and elapsed time. Applying the estimate to short segments we deduce directly that the maximum curvature decays exponentially to 1. This gives a self-contained proof of Grayson's theorem which does not require the monotonicity formula or the classification of singularities.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c0937ab1efd20b5ab783b856837521f6b63327f2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Lami Kim
2. Yejuan Peng
3. Qing Cheng
4. Yoshihiro Tonegawa
5. Hongcang Yang
6. Kota Kasai

In Geometry and Topology

1. Lami Kim
2. Yejuan Peng
3. Qing Cheng
4. Yoshihiro Tonegawa

Citations in the corpus

• On the mean curvature flow of grain boundaries (2015, result): “For n = 1, it is well known that the curves stay embedded until they become convex and shrink to a point by the results due to Gage-Hamilton [22] and Grayson [25] (see also [4] for the elegant and short proof).”
• Complete self-shrinkers of the mean curvature flow (2012, background): “According to the classification of minimal surface in sphere Sp+1( √ 2) with constant Gauss curvature due to Bryant [3] (cf. Calabi [4], Kenmotsu [17] and Wallach [23]), we know that M2 is isometric to a Boruvka sphere S2( √ m(m+ 1)) in S2m( √ 2) with p = 2m− 1 and |A|2 = 2− 2 m(m+1) or a compact fl”