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Proceedings of the Conference on Elliptic and Parabolic Equations, held at Gregynog, August 1989.
 

Summary: Proceedings of the Conference on Elliptic and Parabolic Equations,
held at Gregynog, August 1989.
Shrinking Doughnuts.
SIGURD B. ANGENENT.
Introduction.
Let Mn
be a smooth compact oriented manifold, and let X : M
[0, T ) Rn+1
be a smooth family of immersions of M in n+1 dimensional
Euclidean space. The orientation of M allows one to define a unique smooth
unit normal vector field X : M [0, T ) Rn+1
. Given this choice of X,
we can define the principal curvatures, 1, . . . , n, of the immersion X(, t)
and the mean curvature HX = (1 + . . . + n)/n in the usual way. By
definition, the family of immersions X(, t) "moves by its mean curvature"
if the normal velocity satisfies
(1) Xt(p, t), X (p, t) = nHX(p, t)
for all (p, t) M [0, T ). Here x, y = x0y0 + + xnyn denotes the
Euclidean inner product on Rn+1
.

  

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

 

Collections: Mathematics