Levi equation for almost complex structures

Abstract

Let (R, J) be the space R equipped with an almost complex structure J . We recall that J is a differentiable map R → GL(4,R) such that J(p) = −Id, for every p ∈ R. Let M be a differentiable hypersurface in R. For every p ∈ M the tangent hyperplane TpM contains a (unique) J-invariant plane T J p M . The distribution of planes p → T J p M is called the Levi distribution on M , and M is said to be J-Levi flat whenever p → T J p M is integrable. In view of Frobenius Theorem M is then foliated by regular surfaces on which J induces an integrable almost complex structure. Consequently M is foliated by complex curves, whose complex structure is in general different from that induced by J0, the standard one. Let J = J0. The problem of finding a Levi flat hypersurface with a prescribed boundary Γ has been extensively studied by methods of the geometric theory of several complex variables (cf. [BG], [BK], [A], [S], [K], [CS], [ST]). A different approach is found in [SlT] where the boundary problem for Levi flat graphs is reduced to a Dirichlet problem for a nonlinear, second order, elliptic degenerate operator L, the so called Levi operator (see (??)