Annamaria Montanari

Maximum and Comparison Principles for Convex Functions on the Heisenberg Group

Abstract We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge–Ampère type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge–Ampère measures for convex functions in the Heisenberg group.

Smoothness of Lipschitz-continuous graphs with nonvanishing Levi curvature

In (1), (2), ~=(x , y, t) denotes the point of R 3, ut is the first derivative of u with respect to t, and analogous notations are used for the other firstand second-order derivatives of u. The notion of Levi curvature for a real manifold was introduced by E.E. Levi in 1909 in order to characterize the holomorphy domains of C 2. Since then, it has played a crucial role in the geometric theory of several complex variables. In looking for the polynomial hull of a graph, Slodkowski and Tomassini implicitly introduced in 1991 the following definition of Levi curvature for Lipschitz-continuous graphs [16].

Nonsmooth Hörmander vector fields and their control balls

We prove a ball-box theorem for nonsmooth Hörmander vector fields of step s ≥ 2.

REAL HYPERSURFACES EVOLVING BY LEVI CURVATURE: SMOOTH REGULARITY OF SOLUTIONS TO THE PARABOLIC LEVI EQUATION

We prove, with a real analysis technique, the smooth regularity of classical solutions to a nonlinear degenerate parabolic PDE with initial data C 2,α. This equation arises in the study of the geometric properties of the motion by the trace of the Levi form of a real hypersurface in C 2 with Levi curvature different from zero at every point and which is locally the graph of a C 2,α function.

Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems

Abstract We prove integral formulas for closed hypersurfaces in ℂ n+1, which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the “Soap Bubble Theorem” for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.

The Levi Monge-Ampère equation: Smooth regularity of strictly Levi convex solutions

We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is fully non linear and naturally arises in the study of real hypersurfaces in ℂn+1, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real hypersurface ℂn+1 and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface.

Almost Exponential Maps and Integrability Results for a Class of Horizontally Regular Vector Fields

We consider a family

Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations

{\mathcal{H}}:= \{X_1, \dots, X_m\}

A Frobenius-type theorem for singular Lipschitz distributions

of C1 vector fields in ℝn and we discuss the associated

Sobolev and Morrey Estimates for Non-Smooth Vector Fields of Step Two

{\mathcal{H}}