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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
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[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
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