Published Paper
(2002)
Author:
Carlo Mantegazza
Journal: GAFA [MathSciNet]
Volume: 12
Pages: 138-182
Abstract:
We consider the gradient flow associated to the
following functionals
|
Fm(j) = |
ó
õ
|
M
|
1+|Ñmn|2 dm . |
|
The functionals are defined on hypersurfaces immersed in Rn+1 via
a map j:M® Rn+1, where M is a smooth closed and
connected n-dimensional manifold without boundary.
Here m and Ñ are respectively the canonical measure and
the Levi-Civita connection on the Riemannian manifold (M,g), where
the metric g is obtained by pulling back on M the usual metric of
Rn+1 with the map j. The symbol Ñm denotes the
m-th iterated covariant derivative and n is a unit normal local
vector field to the hypersurface.
Our main result is that if the order of derivation m Î N is
strictly larger than the integer part of n/2 then singularities in
finite time cannot occur during the evolution.
These geometric functionals are related to similar ones proposed by
Ennio De Giorgi, who conjectured for them an analogous regularity
result. In the final section we discuss the original conjecture
of De Giorgi and some related problems.
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[BibTeX Entry]
Available Files:
geoevol.ps (geoevol.pdf)
abstract.tex (abstract.ps, abstract.pdf)
