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The Monge problem of “piles and holes” on the torus and the problem of small denominators. (English. Russian original) Zbl 1412.57024

Sib. Math. J. 59, No. 6, 1090-1093 (2018); translation from Sib. Mat. Zh. 59, No. 6, 1370-1374 (2018).
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 [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)].

MSC:

57R50 Differential topological aspects of diffeomorphisms
57R35 Differentiable mappings in differential topology

Citations:

Zbl 0135.42701
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Full Text: DOI

References:

[1] Bogachev V. I. and Kolesnikov A. V., “The Monge-Kantorovich problem: achievements, connections, and perspectives,” Russian Math. Surveys, vol. 67, No. 5, 785-890 (2012). · Zbl 1276.28029 · doi:10.1070/RM2012v067n05ABEH004808
[2] Bogachev V. I., Fundamentals of Measure Theory. Vol. 2 [Russian], Nauchno-Issled. Tsentr “Regulyarnaya i Khaoticheskaya Dinamika,” Moscow and Izhevsk (2003).
[3] Moser J., “On the volume elements on a manifold,” Trans. Amer. Math. Soc., vol. 120, No. 2, 286-294 (1965). · Zbl 0141.19407 · doi:10.1090/S0002-9947-1965-0182927-5
[4] Banyaga A., “Formes-volume sur les variétés ‘a bord,” Enseign. Math., vol. 20, No. 2, 127-131 (1974). · Zbl 0281.58001
[5] Greene R. E. and Shiohama K., “Diffeomorphisms and volume-preserving embeddings of noncompact manifolds,” Trans. Amer. Math. Soc., vol. 255, 403-414 (1979). · Zbl 0418.58002 · doi:10.1090/S0002-9947-1979-0542888-3
[6] Yagasaki T., “Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends,” Trans. Amer. Math. Soc., vol. 362, No. 11, 5745-5770 (2010). · Zbl 1206.57041 · doi:10.1090/S0002-9947-2010-05101-3
[7] Arnold V. I., “Small denominators and problems of stability of motion in classical and celestial mechanics,” Russian Math. Surveys, vol. 18, No. 6, 85-191 (1963). · Zbl 0135.42701 · doi:10.1070/RM1963v018n06ABEH001143
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