Abstract
We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is fully non linear and naturally arises in the study of real hypersurfaces in ℂn+1, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real hypersurface ℂn+1 and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface.
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References
Bedford, E. and Gaveau, B. Hypersurfaces with bounded Levi form,Indiana Univ. J.,27(5), 867–873, (1978).
Caffarelli, L., Kohn, J.J., Niremberg, L., and Spruck, J. The Dirichlet problem for non-linear second-order elliptic equations II: Complex Monge-Ampère and uniformly elliptic equations,Comm. Pure Appl. Math.,38, 209–252, (1985).
Capogna, L., Danielli, D., and Garofalo, N. Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations,Am. J. Math.,118(6), 1153–1196, (1996).
Citti, G. C∞ regularity of solutions of a quasilinear equation related to the Levi operator,Ann. Scuola Norm. Sup. di Pisa Cl. Sci.,4, vol. XXIII, 483–529, (1996).
Citti, G. C∞ regularity of solutions of the Levi equation,Ann. Inst. H. Poincaré, Anal. non Linéaire,15(4), 517–534, (1998).
Citti, G., Lanconelli, E., and Montanari, A. On the smoothness of viscosity solutions of the prescribed Levi-curvature equation,Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.,10, 61–68, (1999).
Citti, G., Lanconelli, E., and Montanari, A. Smoothness of Lipschitz continuous graphs with non vanishing Levi curvature,Acta Math.,188, 87–128, (2002).
Citti, G. and Montanari, A. Strong solutions for the Levi curvature equation,Adv. Dig. Equ.,5(1–3), 323–342, (2000).
Citti, G. and Montanari, A. Regularity properties of Levi flat graphs,C.R. Acad. Sci. Paris,329(1), 1049–1054, (1999).
Citti, G. and Montanari, A. Analytic estimates for solutions of the Levi equation,J. Diff. Equ.,173, 356–389, (2001).
Citti, G. and Montanari, A. C∞ regularity of solutions of an equation of Levi’s type in ℝ2n+1,Ann. Mat. Pura Appl.,180, 27–58, (2001).
Citti, G. and Montanari, A. Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations,Trans. Am. Math. Soc.,354, 2819–2848, (2002).
D’Angelo, J.P.Several Complex Variables and the Geometry of Real Hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, (1993).
Folland, G.B. Subelliptic estimates and functions spaces on nilpotent Lie groups,Ark. Mat.,13, 161–207, (1975).
Folland, G.B. and Stein, E.M. Estimates for the\(\bar \partial _b \) complex and analysis on the Heisenberg group,Comm. Pure Appl. Math.,20, 429–522, (1974).
Gilgarg, D. and Trudinger, N.S.Elliptic Partial Differential Equations of Second-Order, Grundlehrer der Math. Wiss., vol. 224, Springer-Verlag, New York, (1977).
Hörmander, L.An Introduction to Complex Analysis in Several Variables, Von Nostrand, Princeton, NJ, (1966).
Hörmander, L. Hypoelliptic second-order differential equations,Acta Math.,119, 147–171, (1967).
Krantz, S.Function Theory of Several Complex Variables, John Wiley & Sons, New York, (1982).
Lascialfari, F. and Montanari, A. Smooth regularity for solutions of the Levi Monge-Ampère equation, to appear onAtti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.,12, 115–123, (2001).
Montanari, A. Hölder a priori estimates for second-order tangential operators on CR manifolds,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), vol.II, 345–378, (2003).
Nagel, A., Stein, E.M., and Wainger, S. Balls and metrics defined by vector fields I: basic properties,Acta Math.,155, 103–147, (1985).
Range, R.M.Holomorphic Functions and Integral Representation Formulas in Several Complex Variables, Springer-Verlag, New York, (1986).
Rothschild, L.P. and Stein, E.M. Hypoelliptic differential operators on nilpotent groups,Acta Math.,137, 247–320, (1977).
Sánchez-Calle, A. Fundamental solutions and geometry of the sum of squares of vector fields,Invent. Math.,78, 143–160, (1984).
Slodkowski, Z. and Tomassini, G. The Levi equation in higher dimension and relationships to the envelope of holomorphy,Am. J. Math.,116, 479–499, (1994).
Slodkowski, Z. and Tomassini, G. Weak solutions for the Levi equation and envelope of holomorphy,J. Funct. Anal.,101(4), 392–407, (1991).
Tomassini, G. Geometric Properties of Solutions of the Levi equation,Ann. Mat. Pura Appl. (4),152, 331–344, (1988).
Xu, C.J. Regularity for quasilinear second-order subelliptic equations,Comm. Pure Appl. Math.,45, 77–96, (1992).
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Communicated by Jeffery McNeal
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Montanari, A., Lascialfari, F. The Levi Monge-Ampère equation: Smooth regularity of strictly Levi convex solutions. J Geom Anal 14, 331–353 (2004). https://doi.org/10.1007/BF02922076
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DOI: https://doi.org/10.1007/BF02922076