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SUMMARY:Mean curvature flow for spacelike surfaces\, holomorphic discs and
  the Caratheodory Conjecture. - Wilhelm Klingenberg (Durham)
DTSTART:20100208T160000Z
DTEND:20100208T170000Z
UID:TALK22902@talks.cam.ac.uk
CONTACT:Prof. Neshan Wickramasekera
DESCRIPTION:We outline joint work with Brendan Guilfoyle\, which establish
 es a proof of\nthe Caratheodory Conjecture. This claims that every C3 - di
 fferentiable\nsphere in Euclidean space admits at least two umbilic points
 . (These are\nlocally spherical points\; at such points both principal cur
 vatures are\nequal\, and every tangent vector is a principal direction).\n
 Remark: This is one more umbilic than needs to appear for topological\nrea
 sons\, namely the nonvanishing of the Euler number of the sphere (thereby\
 npresents an instance of rigidity).\nOur proof is inspired by Gromov's sym
 plectic rigidity-flexibility dichotomy\n(specifically by his approach to t
 he rigidity of convex surfaces which lead\nhim to the development of his t
 heory of pseudoholomorphic curves). It uses\nnew a - priory gradient estim
 ates for Mean Curvature Flow in manifolds of\nsplit signature (building on
  work of Bartnik and Ecker-Huisken). The latter\nallows us to construct a 
 holomorphic disc with boundary encircling an\nisolated umbilic point (in a
  symplectic model space). This results in\nsufficient rigidity to prove CC
  in the spirit of said dichotomy.\n
LOCATION:CMS\, MR15
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