Differential geometry of real Monge-Ampère foliations
type: Item
instance of: scholarly article
title: Differential geometry of real Monge-Ampère foliations
zbMATH Open document ID: 0636.53012
DOI: 10.1007/BF01162241
author: Robert L. Foote
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173839
review text: Let u be a real valued function defined on the open set \(\Omega\) in \({\mathbb{R}}^ n,\) and let H(u) be the Hessian of u. Then the Monge-Ampère equation is det H(u)\(=0\). The author proves that if the rank of H(u) is everywhere n-k, \(1\leq k\leq n\), then \(\Omega\) is foliated by k-planes in such a way that u is linear on each leaf. A consequence is a characterization of the Euclidean ball up to affine diffeomorphism which is due originally to \textit{D. Burns} [Ann. Math., II. Ser. 115, 349-373 (1982; Zbl 0507.32011)]. Conditions for the solvability of the Cauchy problem are also obtained, as well as a characterization of algebraic submanifols of \({\mathbb{C}}^ N.\) These results require that the data have only finitely many continuous derivatives, which could not be obtained by previous methods.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 4036619
zbMATH Keywords: foliation by planes
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: characterization of the Euclidean ball
zbMATH Keywords: Cauchy problem
zbMATH Keywords: algebraic submanifols
Wikidata QID: Q115393979
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4036619