JournalsrmiVol. 20, No. 1pp. 151–182

Levi equation for almost complex structures

  • Giovanna Citti

    Università di Bologna, Italy
  • Giuseppe Tomassini

    Scuola Normale Superiore, Pisa, Italy
Levi equation for almost complex structures cover
Download PDF


In this paper we are dealing with the boundary problem for Levi flat graphs in the space R4\mathbb{R}^4, endowed with an almost complex structure JJ. This problem can be formalized as a Dirichlet problem for a quasilinear degenerate elliptic equation, called Levi equation. The Levi equation has the form

D12+D22D1f=0,D_1^2 + D^2_2 - D_1f = 0,

where D1D_1 and D2D_2 are nonlinear vector fields. Under geometrical assumptions on the boundary a lipschitz continuous viscosity solution is found. The regularity of the viscosity solution is studied in suitable anisotropical Sobolev spaces, and it is proved that the solution has derivatives of any order in the direction of the vectors D1D_1 and D2D_2 i.e. it is of class CC^\infty in these directions, but not necessary regular in the third direction of the space. Finally, after proving a weak version of the Frobenius theorem, we show that the graph of the solution is foliated in holomorphic curves.

Cite this article

Giovanna Citti, Giuseppe Tomassini, Levi equation for almost complex structures. Rev. Mat. Iberoam. 20 (2004), no. 1, pp. 151–182

DOI 10.4171/RMI/384