Entire solutions of the degenerateMonge-Ampere equation with a finite\n number of singularities

2016/01/07 by José A. Gálvez, Galvez, Jose' Antonio, Barbara Nelli +1
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1601.01474

Abstract

We determine the global behavior of every C²-solution to the two-dimensional\ndegenerate Monge-Ampere equation, u_xxu_yy-u_xy²=0, over the finitely\npunctured plane. With this, we classify every solution in the once or twice\npunctured plane. Moreover, when we have more than two singularities, if the\nsolution u is not linear in a half-strip, we obtain that the singularities are\nplaced at the vertices of a convex polyhedron P and the graph of u is made by\npieces of cones outside of P which are suitably glued along the sides of the\npolyhedron. Finally, if we look for analytic solutions, then there is at most\none singularity and the graph of u is either a cylinder (no singularity) or a\ncone (one singularity).\n

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