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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., [email protected]
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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

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