With Levi-Monge-Ampère equations (LMA equations) we denote a
class of fully nonlinear PDE's which can be viewed as the
pseudoconvex counterpart of the usual real curvatures
equations (mean curvature equation, Gauss curvature
equations, etc..) and of the Monge-Ampère equations [ML].
They are NONELLIPTIC at any point and they are subelliptic when
computed on functions which are pseudoconex in a suitable
generalized sense. From this crucial property several results
have been already obtained. Our aim is to first prove optimal
regularity of viscosity solutions for the stationary
problem

Let M be a real hypersurface in C^{n+1}, boundary of a domain D.

The Levi form at a point p of M is a Hermitian form on the tangent

space. Since this space has

complex n, the Levi form has n real eigenvalues. Given a

generalized symmetric function ( in the sense of

Caffarelli-Nirenberg-Spruck [CaNiSp]) it is

natural to call S- Levi curvature of M at p, the function S computed

on the eigenvalues of the Levi form. If M is the graph of a function u, and one requires that

M has a given

S-curvature, one obtains a second order PDO which is fully

nonlinear and NOT elliptic at any point. We call this

Levi-Monge-Ampère operator (LMA

operator, in short). The LMA class was implicitly introduced by

Bedford and Gaveau [BeGa] and by Tomassini [To], in studying pseudoconvex domains and

envelopes of holomrphy

in C^{n+1}. The LMA class have been precisely formalized by Montanari

and Lanconelli [MoLa]. In [MoLa] it is proved that the natural

linearizations of the LMA

operators are NOT elliptic at any point. However, they become

subelliptic when computed on the S-pseudoconcex functions. Indeed, they

can be written as "sums of

squares" of NON linear vector fields ( i.e. nonlinear first order

PDE).On S-pseudoconvex functions vector fields directions

are the only ellipticity

directions for the LMA operator. However, the missing ellipticity direction can

be recovered by commutation.

This is the crucial property on which are based most of the results in

literature on the LMA operators. We now summarize the state of the arts on this topic. The following results have already been proved.

(a). The Dirichlet problem for the LMA operators has been studied by Da Lio and Montanari[DaMo]. They prove existence and

uniqueness of a Lipschitz

continuous viscosity solution under quite optimal assumptions. The

problem of the optimal regularity of these solutions is quite

completely open. Only in the lowest

dimensional case a result by Citti-Lanconelli-Montanari [CiLaMo] gives

an almost complete solution of the problem. Indeed, in [CLM] it is proved

that all the locally

Lipschitz viscosity solutions to the Levi equation are of class

C^{infty}if the given curvature is smooth and strictly positive. The

problem, in higher dimension,

is very hard, and cannot be faced by using the usual known

techniques for the fully nonlinear elliptic equations. (b) Strong Comparison Principle, in every

dimension [MoLa]. (c) Spherical symmetry of the bounded Reinhardt

domains of C^2 with

constant Levi curvature [HoLa]. In

[MaMo2] some integral formulas, and isoperimetric inequalities for

compact domains,

involving the Levi curvatures are proved. The same paper contains some

spherical symmetry result for starlike domains with constant Levi

curvatures. In [MaMo3] by using differential geometry tecniques, we prove that if the unit characteristic direction is a geodesic, then it is an eigenvector of the second fundamental form and the relative eigenvalue is constant. As an application we get a symmetry result, of Alexandrov type, for compact hypersurfaces with positive constant Levi mean curvature. In [MoMo] it is proved that the Levi-umbelical surfaces (i.e. with all the eigenvalues of the Levi form equal and constant) are spheres, if compact, cylinders if not bounded. (d) Gradient interior estimates for the Levi-mean curvature equation [MaMo1]. The optimal regularity of

this solutions could be

studied with the techniques introduced in [CiLaMo] for the Levi

equation in the lowest dimension.

We therefore plan to first study regularity

properties of the solutions to fully nonlinear PDE's modeled on Levi equations We then plan to study the motion by Levi curvature and to get from this study new information on symmetry properties. We plan to study some notions of curvatures associated with pseudoconvexity and the Levi form the way the classical Gauss and Mean curvatures are related to the convexity and to the Hessian matrix. In particular, given a prescribed non negative function k, the Levi Monge Ampère equation for the graph of a real value function u is det L = k(x,u)(1+|Du|^2)^ {(n+1)/2}, where L is the Levi form of the graph u and Du is the Euclidean gradient of u. More generally, we shall consider elementary symmetric functions in the eigenvalues of the Levi form L. These curvature equations contain information about the geometric feature of a closed hypersurface and the curvature operator lead to a new class of second order fully nonlinear equations whose characteristic form, when computed on generalized pseudoconvex functions, is nonnegative definite with kernel of dimension one. Thus, the equations are not elliptic at any point. However, they have the following redeeming feature: the missing ellipticity direction can be recovered by suitable commutation relations. We shall use this property to study existence, uniqueness and regularity of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi curvature. To attack the problem of local regularity of Lipschitz continuous viscosity solutions in THE FIRST YEAR we plan to study the regularity theory for three related problems: [1.] The Levi Monge Ampère equation in cylindrical coordinates. In this case we can prove that the prescribed Levi curvature equation is a quasilinear degenerate elliptic equation in two variables, whose prototype is the Grushin operator. [2.] The subelliptic Monge Ampère equation, with respect to left invariant vector fields [3.] Elementary symmetric function in the eigenvalues of real matrices of the type A(x,Du)D^2uA^T(x,Du), where D^2u is the real Hessian, Du is the Euclidean gradient of u, and whose classical solutions are good candidates to approximate viscosity solutions of the previous PDE's. The problem of existence and uniqueness of viscosity for the Dirichlet problem will be jointly studied with Francesca Da Lio, University of Padova. To attack the problem of regularity for these equations we plan to use the regularity theory developed in [CaCa] and in [CaNiSp]. In THE SECOND YEAR we plan to study the related diffusion problems. First of all we plan to study the motion by the trace of the Levi form. A beautiful paper by Huisken and Klingerberg [HK] asserts that if the initial hypersurface is strictly pseudoconvex and C^{2,\alpha} then for small time the evolution by the trace of the Levi form is C^{2,\alpha}. Moreover, by [Mo] the evolution is smooth for small time. We plan to study what happen to Levi flat sides of the initial hypersurface by the Levi curvature flow. We also plan to study existence, uniqueness and regularity for the motion by elementary symmetric functions of the Levi Form and to apply these results to get isoperimetric estimates for the initial problems and characterization properties for closed hypersurfaces with some constant Levi curvatures.

equations. AMS , Providence, RI, 1995.

[CaNiSp] L. Caffarelli, L.Nirenberg,J.Spruck, Comm.Pure Appl. Math. (1985)

[CiLaMo] G.Citti,E.Lanconelli,A.Montatori, Smoothness of Lipschtiz

continuous graphs with nonvanisching Levi curvature, Acta Math., 188,

(2002) 87-128

[DaMo] F. Da Lio, A. Montanari: Existence and Uniqueness of Lipschitz

Continuous Graphs, Ann. Inst. H. Poincaré Anal. Non Linèaire 23 no. 1

(2006) , 1-28.

[HK] G. Huisken; W. Klingenberg. Flow of real hypersurfaces by the trace of the Levi form.

domains of C^2, Contemporary Math, 400 (2006), 129-14

[MaMo1] V. Martino, A. Montanari: Local Lipschitz continuity of graphs

with prescribed Levi urature, NoDEA, Nonlinear Differential Equations and Applications, 14, 2007,377-390 [MaMo2]. V. Martino, A. Montanari: Integral formulas for a class of

curvature PDE's and applications., Forum Mathematicum, Issue 22:2 (2010), 255—267 [MaMo3] V. Martino, A. Montanari,

[MoLa] A. Montanari, E. Lanconelli, Pseudoconvex Fully Nonlinear

Partiall Differential Operators. J.Diff. Eq. 202(2004) 306-333.

[MoMo] R. Monti D.Morbidelli., J. Reine Angew. Math. 603, (2007), 113-131.

[To] G. Tomassini, Ann. Mat. Pura Appl.(1988)

**,**then to study the flow by Levi curvature to get new information about symmetry problems.Let M be a real hypersurface in C^{n+1}, boundary of a domain D.

The Levi form at a point p of M is a Hermitian form on the tangent

space. Since this space has

complex n, the Levi form has n real eigenvalues. Given a

generalized symmetric function ( in the sense of

Caffarelli-Nirenberg-Spruck [CaNiSp]) it is

natural to call S- Levi curvature of M at p, the function S computed

on the eigenvalues of the Levi form. If M is the graph of a function u, and one requires that

M has a given

S-curvature, one obtains a second order PDO which is fully

nonlinear and NOT elliptic at any point. We call this

Levi-Monge-Ampère operator (LMA

operator, in short). The LMA class was implicitly introduced by

Bedford and Gaveau [BeGa] and by Tomassini [To], in studying pseudoconvex domains and

envelopes of holomrphy

in C^{n+1}. The LMA class have been precisely formalized by Montanari

and Lanconelli [MoLa]. In [MoLa] it is proved that the natural

linearizations of the LMA

operators are NOT elliptic at any point. However, they become

subelliptic when computed on the S-pseudoconcex functions. Indeed, they

can be written as "sums of

squares" of NON linear vector fields ( i.e. nonlinear first order

PDE).On S-pseudoconvex functions vector fields directions

are the only ellipticity

directions for the LMA operator. However, the missing ellipticity direction can

be recovered by commutation.

This is the crucial property on which are based most of the results in

literature on the LMA operators. We now summarize the state of the arts on this topic. The following results have already been proved.

(a). The Dirichlet problem for the LMA operators has been studied by Da Lio and Montanari[DaMo]. They prove existence and

uniqueness of a Lipschitz

continuous viscosity solution under quite optimal assumptions. The

problem of the optimal regularity of these solutions is quite

completely open. Only in the lowest

dimensional case a result by Citti-Lanconelli-Montanari [CiLaMo] gives

an almost complete solution of the problem. Indeed, in [CLM] it is proved

that all the locally

Lipschitz viscosity solutions to the Levi equation are of class

C^{infty}if the given curvature is smooth and strictly positive. The

problem, in higher dimension,

is very hard, and cannot be faced by using the usual known

techniques for the fully nonlinear elliptic equations. (b) Strong Comparison Principle, in every

dimension [MoLa]. (c) Spherical symmetry of the bounded Reinhardt

domains of C^2 with

constant Levi curvature [HoLa]. In

[MaMo2] some integral formulas, and isoperimetric inequalities for

compact domains,

involving the Levi curvatures are proved. The same paper contains some

spherical symmetry result for starlike domains with constant Levi

curvatures. In [MaMo3] by using differential geometry tecniques, we prove that if the unit characteristic direction is a geodesic, then it is an eigenvector of the second fundamental form and the relative eigenvalue is constant. As an application we get a symmetry result, of Alexandrov type, for compact hypersurfaces with positive constant Levi mean curvature. In [MoMo] it is proved that the Levi-umbelical surfaces (i.e. with all the eigenvalues of the Levi form equal and constant) are spheres, if compact, cylinders if not bounded. (d) Gradient interior estimates for the Levi-mean curvature equation [MaMo1]. The optimal regularity of

this solutions could be

studied with the techniques introduced in [CiLaMo] for the Levi

equation in the lowest dimension.

We therefore plan to first study regularity

properties of the solutions to fully nonlinear PDE's modeled on Levi equations We then plan to study the motion by Levi curvature and to get from this study new information on symmetry properties. We plan to study some notions of curvatures associated with pseudoconvexity and the Levi form the way the classical Gauss and Mean curvatures are related to the convexity and to the Hessian matrix. In particular, given a prescribed non negative function k, the Levi Monge Ampère equation for the graph of a real value function u is det L = k(x,u)(1+|Du|^2)^ {(n+1)/2}, where L is the Levi form of the graph u and Du is the Euclidean gradient of u. More generally, we shall consider elementary symmetric functions in the eigenvalues of the Levi form L. These curvature equations contain information about the geometric feature of a closed hypersurface and the curvature operator lead to a new class of second order fully nonlinear equations whose characteristic form, when computed on generalized pseudoconvex functions, is nonnegative definite with kernel of dimension one. Thus, the equations are not elliptic at any point. However, they have the following redeeming feature: the missing ellipticity direction can be recovered by suitable commutation relations. We shall use this property to study existence, uniqueness and regularity of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi curvature. To attack the problem of local regularity of Lipschitz continuous viscosity solutions in THE FIRST YEAR we plan to study the regularity theory for three related problems: [1.] The Levi Monge Ampère equation in cylindrical coordinates. In this case we can prove that the prescribed Levi curvature equation is a quasilinear degenerate elliptic equation in two variables, whose prototype is the Grushin operator. [2.] The subelliptic Monge Ampère equation, with respect to left invariant vector fields [3.] Elementary symmetric function in the eigenvalues of real matrices of the type A(x,Du)D^2uA^T(x,Du), where D^2u is the real Hessian, Du is the Euclidean gradient of u, and whose classical solutions are good candidates to approximate viscosity solutions of the previous PDE's. The problem of existence and uniqueness of viscosity for the Dirichlet problem will be jointly studied with Francesca Da Lio, University of Padova. To attack the problem of regularity for these equations we plan to use the regularity theory developed in [CaCa] and in [CaNiSp]. In THE SECOND YEAR we plan to study the related diffusion problems. First of all we plan to study the motion by the trace of the Levi form. A beautiful paper by Huisken and Klingerberg [HK] asserts that if the initial hypersurface is strictly pseudoconvex and C^{2,\alpha} then for small time the evolution by the trace of the Levi form is C^{2,\alpha}. Moreover, by [Mo] the evolution is smooth for small time. We plan to study what happen to Levi flat sides of the initial hypersurface by the Levi curvature flow. We also plan to study existence, uniqueness and regularity for the motion by elementary symmetric functions of the Levi Form and to apply these results to get isoperimetric estimates for the initial problems and characterization properties for closed hypersurfaces with some constant Levi curvatures.

**References**[CaCa] Caffarelli, Luis A.; X Cabré, Xavier Fully nonlinear ellipticequations. AMS , Providence, RI, 1995.

[CaNiSp] L. Caffarelli, L.Nirenberg,J.Spruck, Comm.Pure Appl. Math. (1985)

[CiLaMo] G.Citti,E.Lanconelli,A.Montatori, Smoothness of Lipschtiz

continuous graphs with nonvanisching Levi curvature, Acta Math., 188,

(2002) 87-128

[DaMo] F. Da Lio, A. Montanari: Existence and Uniqueness of Lipschitz

Continuous Graphs, Ann. Inst. H. Poincaré Anal. Non Linèaire 23 no. 1

(2006) , 1-28.

[HK] G. Huisken; W. Klingenberg. Flow of real hypersurfaces by the trace of the Levi form.

*Math. Res. Lett.*6 (1999), no. 5-6, 645--661. [HoLa] J.Hounie.E. Lanconelli, An Alexander type Theorem for Reinhardtdomains of C^2, Contemporary Math, 400 (2006), 129-14

[MaMo1] V. Martino, A. Montanari: Local Lipschitz continuity of graphs

with prescribed Levi urature, NoDEA, Nonlinear Differential Equations and Applications, 14, 2007,377-390 [MaMo2]. V. Martino, A. Montanari: Integral formulas for a class of

curvature PDE's and applications., Forum Mathematicum, Issue 22:2 (2010), 255—267 [MaMo3] V. Martino, A. Montanari,

*On the characteristic direction of real hypersurfaces in***C***N+1 and a symmetry result*, to appear in Advances in Geometry [Mo] A. Montanari, Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation, Comm. Partial Differential Equations 26 (2001), no. 9-10, 1633--1664[MoLa] A. Montanari, E. Lanconelli, Pseudoconvex Fully Nonlinear

Partiall Differential Operators. J.Diff. Eq. 202(2004) 306-333.

[MoMo] R. Monti D.Morbidelli., J. Reine Angew. Math. 603, (2007), 113-131.

[To] G. Tomassini, Ann. Mat. Pura Appl.(1988)