Analyse et géométrie complexes, Géométrie et Topologie
Directed immersions for complex structures
Comptes Rendus. Mathématique, Tome 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 $\left(X,J,g\right)$, and for $ϵ>0$, there exists a smooth function $f:X\to ℝ$ and almost complex structure ${J}^{\prime }$ on $X$ such that $J$ and ${J}^{\prime }$ are ${C}^{0}$-close on the graph of $f$ with respect to the extended metric on $X×ℝ$, and such that the Nijenhuis tensor of ${J}^{\prime }$ 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 : https://doi.org/10.5802/crmath.221
Tobias Shin 1

1. Department of Mathematics, SUNY Stony Brook, 100 Nicolls Rd., Stony Brook, NY 11794, USA
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Tobias Shin. Directed immersions for complex structures. Comptes Rendus. Mathématique, Tome 359 (2021) no. 7, pp. 773-793. doi : 10.5802/crmath.221. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.221/

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