Abstract.
This paper generalizes results of Lempert and Szöke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C 3 everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampère equation. These results are proved using techniques from contact geometry.
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Received: 6 April 2001 / Published online: 2 December 2002
Mathematics Subject Classification (2000): 53C56, 32F, 53C60
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Duchamp, T., Kalka, M. Singular Monge-Ampère foliations. Math. Ann. 325, 187–209 (2003). https://doi.org/10.1007/s00208-002-0378-5
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DOI: https://doi.org/10.1007/s00208-002-0378-5