Differential geometry of real Monge-Ampère foliations
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.
scientific article; zbMATH DE number 4036619
scientific article