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

Jose A. Galvez·Antonio Martinez·Pablo Mira
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Abstract

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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Jose A. Galvez, Antonio Martinez, Pablo Mira. 2004-11-15. The space of solutions to the Hessian one equation in the finitely punctured plane. https://arxiv.org/abs/math/0411325

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