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Inflection Points, Extatic Points and Curve Shortening1

Summary: Inflection Points, Extatic Points
and Curve Shortening1
Sigurd Angenent
Department of Mathematics, UW­Madison
If (M2
, g) is a surface with Riemannian metric, then a family of immersed
curves {Ct | 0 t < T} on M2
evolves by Curve Shortening if
= g, (1)
where g is the geodesic curvature, and is a unit normal to the curve. Since
g can be written as 2
s2 , where s is arclength along C, (1) is essentially a
parabolic equation, i.e. a nonlinear heat equation.
In [An2] it is shown that for any solution {Ct | 0 t < T} of (1) there
is only a discrete set of times at which the immersed curve Ct will have self
tangencies. Hence the number of self- intersections of Ct is always finite, and
it was also shown in [An2] that this number decreases whenever the curve


Source: Angenent, Sigurd - Department of Mathematics, University of Wisconsin at Madison


Collections: Mathematics