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VERSION:2.0
PRODID:-//Memento EPFL//
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SUMMARY:Total curvature of submanifolds
DTSTART:20141210T163000
DTEND:20141210T173000
DTSTAMP:20260428T111501Z
UID:0df057b2a89c527dbfb8639b380befaa7539859bc53fcbdd0b868080
CATEGORIES:Conferences - Seminars
DESCRIPTION:Wolfgang Kühnel (Stuttgart)\nGeometry and Dynamics SeminarAbs
 tract: The classical Gauss-Bonnet theorem states that the total (Gaussian)
  curvature of a compact surface equals its Euler characteristic. The Cohn-
 Vossen inequality on the total curvature gives an estimate for the Gauss-B
 onnet defect that arises for complete\, non-compact surfaces. The nature o
 f this defect was further studied by Shiohama\, Wintgen and others. For mi
 nimal surfaces it is known to be `quantized' by integer multiples of $2\\p
 i$.\nIn higher dimensions this inequality is no longer valid. We present a
  simple 4-dimensional counterexample. Several sufficient intrinsic or extr
 insic criteria are known that imply a vanishing defect. For odd-dimensiona
 l hypersurfaces the situation is completely different since the total curv
 ature fails to be a topological invariant even in the compact case. Instea
 d it depends on the regular homotopy class of the immersion. Here the simp
 lest case is the Whitney-Graustein theorem on closed curves.\nIn the talk 
 we study the difference term for even-dimensional complete submanifolds of
  Euclidean spaces of an appropriate type. It turns out that the Gauss-Bonn
 et defect is an intrinsic quantity of the `set at infinity' or the `ideal 
 boundary'. This allows to discuss the condition for the total curvature to
  be stationary and a possible quantization of the total curvature in this 
 case.
LOCATION:GR A3 30
STATUS:CONFIRMED
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