Complex analysis and geometry, Geometry and Topology
Directed immersions for complex structures
Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 773-793.

We analyze the differential relation corresponding to integrability of almost complex structures, reformulated as a directed immersion relation by Demailly and Gaussier. Using recent results of Clemente from 2020 in combination with this analysis, we show the following two statements: first, there are no formal obstructions to integrability of a complex structure, in the sense of h-principle. Second, for an almost complex manifold with arbitrary metric (X,J,g), and for ϵ>0, there exists a smooth function f:X and almost complex structure J on X such that J and J are C 0 -close on the graph of f with respect to the extended metric on X×, and such that the Nijenhuis tensor of J on the graph has pointwise sup norm less than Cϵ, where C is a constant depending only on J and g.

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DOI: 10.5802/crmath.221
Shin, Tobias 1

1 Department of Mathematics, SUNY Stony Brook, 100 Nicolls Rd., Stony Brook, NY 11794, USA
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Shin, Tobias. Directed immersions for complex structures. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 773-793. doi : 10.5802/crmath.221. http://www.numdam.org/articles/10.5802/crmath.221/

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