Estimates of the Kobayashi-Royden metric in almost complex manifolds. (English) Zbl 1083.32011
A lower estimate is obtained for the Kobayashi-Royden infinitesimal metric on an almost complex manifold \((M,J)\): if \(\rho\) is a \(C^2\) defining function of a relatively compact domain \(D= \{\rho< 0\}\), strictly \(J\)-plurisubharmonic near \(\overline D\), then
\[
K_{(D,J)}(p,v)\geq c\Biggl[{|\partial_J\rho(p)(v- iJ(p)v)|^2\over |\rho(p)|^2}+ {\| v\|\over|\rho(p)}\Biggr]^{1/2}
\]
for all \(p\in D\) and \(v\in T_pM\).
As a consequence, every point of \(M\) has a basis of complete hyperbolic neighbourhoods.
As a consequence, every point of \(M\) has a basis of complete hyperbolic neighbourhoods.
Reviewer: Alexandr Yu. Rashkovsky (Stavanger)
MSC:
| 32F45 | Invariant metrics and pseudodistances in several complex variables |
| 32V40 | Real submanifolds in complex manifolds |
| 32V15 | CR manifolds as boundaries of domains |
| 32H40 | Boundary regularity of mappings in several complex variables |
| 32T15 | Strongly pseudoconvex domains |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
References:
| [1] | Berteloot (F.) -Attraction des disques analytiques et continuité höldé-rienne d’applications holomorphes propres, in Topics in complex analysis (Warsaw, 1992), Banach Center Publ., vol. 31, Polish Acad. Sci., Warsaw, 1995, pp. 91-98. · Zbl 0831.32012 |
| [2] | , Principe de Bloch et estimations de la métrique de Kobayashi des domaines de C 2 , J. Geom. Anal., t. 13 (2003), pp. 29-37. · Zbl 1040.32011 |
| [3] | Chirka (E.M.), Coupet (B.) & Sukhov (A.B.) -On boundary regu-larity of analytic discs, Michigan Math. J., t. 46 (1999), pp. 271-279. · Zbl 0985.32009 |
| [4] | Debalme (R.) -Kobayashi hyperbolicity of almost complex manifolds, Preprint IRMA, arXiv: math.CV/9805130, 1999. |
| [5] | Debalme (R.) & Ivashkovich (S.) -Complete hyperbolic neighborhoods in almost-complex surfaces, Internat. J. Math., t. 12 (2001), pp. 211-221. · Zbl 1110.32306 |
| [6] | Diederich (K.) & Fornaess (J.E.) -Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary, Ann. of Math. (2), t. 110 (1979), pp. 575-592. · Zbl 0394.32012 |
| [7] | Graham (I.) -Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C n with smooth boundary, Trans. Amer. Math. Soc., t. 207 (1975), pp. 219-240. · Zbl 0305.32011 |
| [8] | Haggui (F.) -Fonctions FSH sur une variété presque complexe, C. R. Math. Acad. Sci. Paris, t. 335 (2002), pp. 509-514. · Zbl 1013.32019 |
| [9] | Ivashkovich (S.) & Rosay (J.-P.) -Schwarz-type lemmas for solutions of∂-inequalities and complete hyperbolicity of almost complex manifolds, arXiv: math.CV/0310474, 2003. |
| [10] | Kerzman (N.) & Rosay (J.-P.) -Fonctions plurisousharmoniques d’exhaustion bornées et domaines taut, Math. Ann., t. 257 (1981), pp. 171-184. · Zbl 0451.32012 |
| [11] | Kobayashi (S.) -Almost complex manifolds and hyperbolicity, Results Math., t. 40 (2001), pp. 246-256, Dedicated to Shiing-Shen Chern on his 90th birthday. · Zbl 1004.53052 |
| [12] | Kruglikov (B.S.) -Existence of close pseudoholomorphic disks for al-most complex manifolds and their application to the Kobayashi-Royden pseudonorm, Funktsional. Anal. i Prilozhen., t. 33 (1999), pp. 46-58, 96. · Zbl 0967.32024 |
| [13] | Nijenhuis (A.) & Woolf (W.B.) -Some integration problems in almost-complex and complex manifolds, Ann. of Math. (2), t. 77 (1963), pp. 424-489. · Zbl 0115.16103 |
| [14] | Pinchuk (S.) -The scaling method and holomorphic mappings, in Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 151-161. · Zbl 0744.32013 |
| [15] | Sibony (N.) -A class of hyperbolic manifolds, in Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N.J., 1979), Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 357-372. · Zbl 0476.32033 |
| [16] | Sikorav (J.-C.) -Some properties of holomorphic curves in almost com-plex manifolds, in Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 165-189. · Zbl 0802.53001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
