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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.
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