ICTP /SISSA Joint Colloquium in Mathematics - 7 July 2009

[bergamo]

ICTP/SISSA JOINT COLLOQUIUM IN MATHEMATICS

ANNOUNCEMENT

Tuesday, 7 July at 14:00 hrs.

Professor Ermanno Lanconelli
(Universita' di Bologna, Italy)

"Levi curvatures and related problems"

Abstract:
The Levi curvatures of a real hypersurfaces of C^{n+1} are related to 
the Levi form the way the usual Mean or Gauss curvatures are related to 
the real Hessian form.
They were implicitly introduced by Bedford and Gaveau in 1978, and more 
explicitly, in  the case n=1, by Tomassini in 1988.
Later on, Montanari and Lanconelli extended these notions to any 
dimension, following the  lines of the works by Caffarelli, Nirenberg 
and Spruck on the real Monge-Ampere equations. Montanari and Lanconelli 
showed that, for hypersurface graphs on real functions, the Levi 
curvatures can be expressed in terms of second order fully nonlinear 
PDO's with an underlying sub-Riemannian structure.  The properties of 
these structures allowed one to obtain:
1) A strong comparison principle for hypersurfaces having the same Levi 
curvatures;
2)  Isoperimetric inequalities;
3) Alexandrov-type sphere theorems;
4) Existence theorems of Lipschitz-continuous viscosity solutions to the 
prescribed  Levi-curvature problems
5)  (only for n=1) Smoothness of the Lipschitz-continuous viscosity 
solutions.
The results in 3) and 5) only give partial answers to problems which are 
still largely open.

Venue:  SEMINAR ROOM, (ICTP Leonardo Building, first floor)