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Symplectic structures and symmetries of solutions of the complex Monge-Ampére equation

Published online by Cambridge University Press:  22 January 2016

Stanley M. Einstein-Matthews
Stanley M. Einstein-Matthews*
Affiliation:
Department of Mathematics, Howard University, 2441 6th Street, N.W. Washington, D.C 20059, U.S.A., smem@scs.howard.edu
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Abstract.

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The graphs that arise from the gradients of solutions u of the homogeneous complex Monge-Ampère equation are characterized in terms of the natural symplectic structure on the cotangent bundle. This characterization is invariant under symplectic biholomorphisms. Using the symplectic structures we construct symmetries (to be called Lempert transformations) for real valued functions u which are absolutely continuous on lines. We then use these symmetries to generate interesting solutions to the homogeneous complex Monge-Ampère equation and to transform the Poincaré-Lelong equation and the ∂-equation. An example of Lempert transform is given and the main theorem is applied to prove regularity results for exterior nonlinear Dirichlet problem for the homogeneous complex Monge-Ampère equation.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 150 , June 1998 , pp. 63 - 83
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

References

[An.1] Arnold, V. I., Mathematical Methods of Classical Mechanics, Graduate Texts in Math., 60 (1978).Google Scholar
[B-T.1] Bedford, E. and Taylor, B. A., The Dirichlet Problem for a Complex Monge-Ampère Equation, Inventiones Math., 37 (1976), 1–44.Google Scholar
[B-T.2] Bedford, E. and Taylor, B. A., A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1–40.CrossRefGoogle Scholar
[Bo.1] Boggess, A., CR Manifolds and the Tangential Cauchy-Riemann Complex, CRS Press., 1991.Google Scholar
[CHL.1] Chin-Huei, Chang, Hu, M. C. and Lee, Hsuan-Pei, Extremal Analytic Discs with Prescribed Boundary Data, Trans. Amer. Math. Soc, 310 (1988), 355–369.Google Scholar
[De.1] Demailly, J.-P., Monge-Ampère Operators, Lelong Numbers and Intersection Theory, Prépublication de l’Institut Fourier, Grenoble, no 173 (1991).Google Scholar
[Kl.1] Klimek, M., Pluripotential Theory, Oxford Science Publications, 1991.Google Scholar
[Le.1] Lelong, P., Fonctions Plurisousharmoniques et Formes Differentielles Positives, Gordon and Breach (1968).Google Scholar
[Lem.1] Lempert, L., La métrique de Kobayashi et la représentation de domaines sur la boule, Bull. Soc. Math. France, 109 (1981), 427–474.Google Scholar
[Lem.2] Lempert, L., Symmetries and other Transformations of the Complex Monge-Ampère Equation, Duke Mathematical Journal, 52–4 (1985), 869–885.Google Scholar
[Lem.3] Lempert, L., Solving the degenerate complex Monge-Ampère equation with one concentrated singularity, Math. Ann., 263 (1983), 515–532.Google Scholar
[Lem.4] Lempert, L., Intrinsic distances and holomorphic retracts, Proc. Conf. Varna 1981 (1984); Complex Analysis and Appl. Sofia, 341–364.Google Scholar
[Lem.5] Lempert, L., Holomorphic retracts and intrinsic metrics in convex domains, Analysis Math., 8 (1982), 257–261.Google Scholar
[Se.1] Semmes, S., Complex Monge-Ampère and Symplectic Manifolds, American Journal of Mathematics, 114 (1990), 495–550.CrossRefGoogle Scholar
[Ri.1] Richman, S., Quasiregular Mappings, Springer-Verlag, 1993.Google Scholar
[Tel.1] Teleman, N., The index of the signature operator on Lipschitz manifolds, Publ. Math. Inst. Hautes Etudes Sci., 58 (1983), 39–78.Google Scholar
[We.1] Weinstein, A., Lectures on Symplectic Manifolds, CBMS Regional Conference Series In Math., 29 (1977).Google Scholar
[Whi.1] Whitney, H., Geometric Integration Theory, Princeton University Press 1957, 1957.Google Scholar