A note on the geometry of pseudoconvex domains of finite type in almost complex manifolds

Bertrand, Florian

doi:10.1090/S0002-9939-2011-11016-5

A note on the geometry of pseudoconvex domains of finite type in almost complex manifolds
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Proc. Amer. Math. Soc. 140 (2012), 1633-1641 Request permission

Abstract:

Let $D=\{\rho <0\}$ be a smooth domain of finite type in an almost complex manifold $(M,J)$ of real dimension four. We assume that the defining function $\rho$ is $J$-plurisubharmonic on a neighborhood of $\overline {D}$. We study the asymptotic behavior of pseudoholomorphic discs contained in the domain $D$.

References

F. Berteloot, Principe de Bloch et estimations de la métrique de Kobayashi des domaines de $\mathbf C^2$, J. Geom. Anal. 13 (2003), no. 1, 29–37 (French, with French summary). MR 1967034, DOI 10.1007/BF02930994
Florian Bertrand, Pseudoconvex regions of finite D’Angelo type in four dimensional almost complex manifolds, Math. Z. 264 (2010), no. 2, 423–457. MR 2574983, DOI 10.1007/s00209-008-0471-x
David W. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), no. 3, 429–466. MR 978601, DOI 10.1007/BF01215657
John P. D’Angelo, Finite type conditions for real hypersurfaces, J. Differential Geometry 14 (1979), no. 1, 59–66 (1980). MR 577878
Hervé Gaussier and Alexandre Sukhov, Estimates of the Kobayashi-Royden metric in almost complex manifolds, Bull. Soc. Math. France 133 (2005), no. 2, 259–273 (English, with English and French summaries). MR 2172267, DOI 10.24033/bsmf.2486
Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $C^{n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219–240. MR 372252, DOI 10.1090/S0002-9947-1975-0372252-8
Sergey Ivashkovich and Jean-Pierre Rosay, Schwarz-type lemmas for solutions of $\overline \partial$-inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 7, 2387–2435 (2005) (English, with English and French summaries). MR 2139698
Albert Nijenhuis and William B. Woolf, Some integration problems in almost-complex and complex manifolds, Ann. of Math. (2) 77 (1963), 424–489. MR 149505, DOI 10.2307/1970126
Jean-Claude Sikorav, Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 165–189. MR 1274929, DOI 10.1007/978-3-0348-8508-9_{6}