# zbMATH — the first resource for mathematics

Geometric properties of solutions of the Levi-equation. (English) Zbl 0681.35017
Given a smooth real hypersurface M in $${\mathbb{C}}^ 2$$ the Levi curvature k is defined. In the case when M is the graph of a function $$u(u_ 1,u_ 2,u_ 3)$$ $$(z_ 1=x_ 1+ix_ 2$$, $$z_ 2=x_ 3+ix_ n)$$ then u satisfies, in terms of k, a second order quasilinear elliptic degenerate equation $$L(k;u)=0$$ called the Levi equation. In the paper geometric properties of smooth solutions of $$L(k;u)=0$$ are discussed.
Reviewer: G.Tomassini

##### MSC:
 35B99 Qualitative properties of solutions to partial differential equations 53B99 Local differential geometry 35J70 Degenerate elliptic equations
Full Text:
##### References:
 [1] Bedford, E.; Gaveau, B., Hypersurfaces with Bounded Levi Form, Ind. Univ. Mat. J., 27, no. 5, 867-873 (1978) · Zbl 0365.32011 [2] Bedford, E.; Gaveau, B., Envelopes of holomorphy of certain 2-spheres inC^2, Am. J. of Math., 105, 975-1009 (1983) · Zbl 0535.32008 [3] Bishop, E., Differentiable manifolds in complex Euclidean space, Duke Math. J., 32, 1-22 (1965) · Zbl 0154.08501 [4] Debiard, A.; Gaveau, B., Problème de Dirichlet pour l’équation de Levi, Bull. Sc. Math., 2a serie, 102, no. 4, 386-369 (1978) · Zbl 0413.35067 [5] Freeman, M., Local complex foliation of real submanifolds, Math. Ann., 209, 1-30 (1974) · Zbl 0267.32006 [6] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Englewood Cliffs, N. J.: Prentice-Hall, Englewood Cliffs, N. J. · Zbl 0144.34903 [7] Gaveau, B., Construction d’enveloppes d’holomorphie, croissance d’ensembles et de fonctions analytiques (1982), Trento: Cours au CIRME, Trento [8] D.Gilbarg - N. S.Trudinger,Elliptic partial differential equations of second order, Grund. der Math. Wiss.224, Springer (1977). · Zbl 0361.35003 [9] Pucci, C., An angle’s maximum principle for the gradient of solutions of elliptic equations, Boll. Un. Mat. Ital., (7), 1-A, 135-139 (1987) · Zbl 0628.35034 [10] Sommer, P., Komplexe-analytische BlÄtterung reeller HyperflÄchen inC^n, Math. Ann., 137, 392-411 (1959) · Zbl 0092.29903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.