The space of solutions to the Hessian one equation in the finitely punctured plane

Details The space of solutions to the Hessian one equation in the finitely punctured plane The space of solutions to the Hessian one equation in the finitely punctured plane

2004-11-15

arxiv.org/abs/math/0411325v1

Mathematics

Analysis of PDEs

14 pages, 3 figures

Scientific paper

We construct the space of solutions to the elliptic Monge-Ampere equation det(D^2 u)=1 in the plane R^2 with n points removed. We show that, modulo equiaffine transformations and for n>1, this space can be seen as an open subset of R^{3n-4}, where the coordinates are described by the conformal equivalence classes of once punctured bounded domains in the complex plane of connectivity n-1. This approach actually provides a constructive procedure that recovers all such solutions to the Monge-Ampere equation, and generalizes a theorem by K. Jorgens.

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