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CATEGORIES:Partial Differential Equations seminar
SUMMARY:A sharp strong maximum principle for singular mini
mal hypersurfaces - Neshan Wickramasekera (Cambrid
ge)
DTSTART;TZID=Europe/London:20130603T150000
DTEND;TZID=Europe/London:20130603T160000
UID:TALK45671AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/45671
DESCRIPTION:If two smooth\, connected\, embedded minimal hyper
surfaces with no\nsingularities satisfy the proper
ty that locally near every common point $p$\, one
hypersurface lies on one side of the other\, then
it is a direct consequence of the Hopf maximum pri
nciple that either the hypersurfaces are disjoint
or they coincide. Given that singularities in mini
mal hypersurfaces are generally unavoidable\, it i
s a natural question to ask if the same result mus
t extend to pairs of singular minimal hypersurface
s (stationary codimesion 1 integral varifolds) wit
h connected supports\; in this case the above ``on
e hypersurface lies locally on one side of the oth
er'' hypothesis can naturally be imposed for each
common point $p$ which is a regular point of at le
ast one hypersurface.\n\nThe answer to this questi
on in general is no in view of simple examples suc
h as two pairs of transversely interecting hyperpl
anes with a common axis. The answer however is yes
if the singular set of one of the hypersurfaces h
as $(n-1)$-dimesional Hausdorff measure zero\, whe
re $n$ is the dimension of the hypersurfaces. I wi
ll discuss this result\, which generalizes and uni
fies previous maximum principles of Ilmanen and So
lomon-White (and thereby unifies all previously kn
own strong maximum principles for singular minimal
hypersurfaces).
LOCATION:CMS\, MR13
CONTACT:Filip Rindler
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