 
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
C
t
= g, (1)
where g is the geodesic curvature, and is a unit normal to the curve. Since
g can be written as 2
C
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
