Ancient solutions to curve shortening with finite total curvature
نویسندگان
چکیده
منابع مشابه
The Total Curvature of a Knotted Curve
1. We are indebted to W. Fenchel [5] for a theorem which is a left closed curve (in ordinary space) has total curvature ≥ 2π. Recently, K. Borsuk [3] gave a new proof of this theorem that applies to curves in R. In a note at the end of this paper, Borsuk asked the question wheter the total curvature of a left knotted curve is always ≥ 4π. The primary purpose of this note is to give an affirmati...
متن کاملCurvature Bound for Curve Shortening Flow via Distance Comparison and a Direct Proof of Grayson’s Theorem
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 se...
متن کاملNon-closed Curves in R with Finite Total First Curvature Arising from the Solutions of an Ode
The solution space of a constant coefficient ODE gives rise to a natural real analytic curve in Euclidean space. We give necessary and sufficient conditions on the ODE to ensure that this curve is a proper embedding of infinite length or has finite total first curvature. If all the roots of the associated characteristic polynomial are simple, we give a uniform upper bound for the total first cu...
متن کاملCurve Shortening and Grayson’s Theorem
In this chapter and the next we discuss the curve shortening flow (CSF). A number of important techniques in the field of geometric flows exhibit themselves in the curve shortening flow in an elegant and less technical way. The CSF was proposed in 1956 by Mullins to model the motion of idealized grain boundaries. In 1978 Brakke studied the mean curvature flow, of which the CSF is the 1-dimensio...
متن کاملImmersed Spheres of Finite Total Curvature into Manifolds
We prove that a sequence of, possibly branched, weak immersions of the two-sphere S into an arbitrary compact riemannian manifold (M, h) with uniformly bounded area and uniformly bounded L−norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of S and whose image is made of a connected union of f...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2020
ISSN: 0002-9947,1088-6850
DOI: 10.1090/tran/8186