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Nearly round spheres look convex

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39 pages
Nearly round spheres look convex Ludovic Rifford Universite de Nice - Sophia Antipolis (Joint work with A. Figalli and C. Villani) Ludovic Rifford Workshop on Optimal transportation and applications

  • nearly round spheres

  • ?? expx

  • expx

  • round sphere

  • injectivity domains

  • geodesic between

  • compact riemannian

  • universite de nice

  • look convex

  • txm ??


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Nearly
(Joint
round
spheres
Ludovic
look
Rifford
Universit´edeNice-SophiaAntipolis
work
with
A.
Ludovic Rifford
Figalli
and
C.
convex
Villani)
Workshop on Optimal transportation and applications
The statement
Theorem (Figalli, R, Villani ’09) Let(M,g)be a C4perturbation of the round sphereSn. Then all injectivity domains of M are uniformly convex.
Ludovic Rifford
Workshop on Optimal transportation and applications
Injectivity domains
Let (M,g) be a smooth compact Riemannian manifold and xM call exponential mapping from Webe fixed.x, the mapping defined as
expx
:TxM v
−→ 7
M expx(v) :=γv(1),
whereγv: [0,1]Mis the unique geodesic starting atxwith speed ˙γv(0) =v.
Ludovic Rifford
Workshop on Optimal transportation and applications