The Monge problem of ``piles and holes'' on the torus and the problem of small denominators
The article discusses the problem of existence of a smooth (analytic) map \(\varphi : M \rightarrow M\) of a closed \(n\)-dimensional manifold \(M\) carrying a differential \(n\)-form into a prescribed volume form (assuming that the integrals of these forms on the whole manifold coincide). For the \(n\)-dimensional torus it is then shown how the problem can be reduced to the problem of small denominators [\textit{V. I. Arnol'd}, Russ. Math. Surv. 18, No. 6, 85--191 (1963; Zbl 0135.42701); translation from Usp. Mat. Nauk 18, No. 6(114), 91--192 (1963)].
scientific article; zbMATH DE number 7035752
scientific article