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CATEGORIES:Cambridge Analysts' Knowledge Exchange
SUMMARY:Bumpy Metrics for Minimal Submanifolds - Paul Mint
er (University of Cambridge)
DTSTART;TZID=Europe/London:20200130T163000
DTEND;TZID=Europe/London:20200130T173000
UID:TALK139075AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/139075
DESCRIPTION:Consider the 2-sphere in Euclidean 3-space with th
e usual round metric\, where we know the geodesics
are arcs of great circles. By rotation we get a 1
-parameter family of geodesics through any given g
eodesic\, which turns out to imply that each geode
sic is degenerate for the length functional since
it then has a non-trivial Jacobi field. However if
we change the metric on the 2-sphere to\, say\, t
hat of a triaxial ellipsoid\, all but 3 of these c
losed geodesics disappear. Perturbing the metric f
urther via adding more "bumps" to the 2-sphere\, a
ll geodesics are in fact non-degenerate. In 1970
Ralph Abraham established that on a compact manifo
ld\, 'almost all' metrics have the property that a
ny geodesic is non-degenerate. This result was the
n extended to the case of minimal submanifolds of
any codimension in 1991 by Brian White to a result
now known as the Bumpy Metrics Theorem. In this t
alk we shall discuss the Bumpy Metrics Theorem\, a
nd then some conjectures we have for extending it
to the case of singular hypersurfaces.
LOCATION:MR14\, Centre for Mathematical Sciences
CONTACT:Renato Velozo
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