Poincaré metric of holomorphic foliations with non-degenerate singularities. (English) Zbl 1527.32017
Summary: Consider a Brody hyperbolic foliation \(\mathcal{F}\) with non-degenerate singularities on a compact complex manifold. We show that its leafwise Poincaré metric is transversally Hölder continuous with a logarithmic slope towards the singular set of \(\mathcal{F}\).
MSC:
| 32M25 | Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions |
| 32S65 | Singularities of holomorphic vector fields and foliations |
References:
| [1] | Dinh, T.-C., Nguyên, V.-A. and Sibony, N., Heat equation and ergodic theorems for Riemann surface laminations, Math. Ann.354(1) (2012) 331-376. · Zbl 1331.37064 |
| [2] | Dinh, T.-C., Nguyên, V.-A. and Sibony, N., Entropy for hyperbolic Riemann surface laminations I, Frontiers in Complex Dynamics: A Volume in Honor of John Milnor’s 80th Birthday, eds. Bonifant, A., Lyubich, M. and Sutherland, S. (Princeton University Press, 2014), pp. 569-592. · Zbl 1352.37145 |
| [3] | Dinh, T.-C., Nguyên, V.-A. and Sibony, N., Entropy for hyperbolic Riemann surface laminations II, Frontiers in Complex Dynamics: A Volume in Honor of John Milnor’s 80th Birthday, eds. Bonifant, A., Lyubich, M. and Sutherland, S. (Princeton University Press, 2014), pp. 593-622. |
| [4] | Dinh, T.-C., Sibony, N., Some open problems on holomorphic foliation theory, Acta Math. Vietnam45(1) (2020) 103-112. · Zbl 1437.37058 |
| [5] | Dulac, H., Recherches sur les points singuliers des équations différentielles, J. l’École Polytech.2(9) (1904) 1-125. · JFM 35.0331.02 |
| [6] | Earle, C. J. and Schatz, A. H., Teichmüller theory for surfaces with boundary, J. Differ. Geom.4 (1970) 169-185. · Zbl 0194.52802 |
| [7] | Fornæss, J. E. and Sibony, N., Riemann surface laminations with singularities, J. Geom. Anal.18(2) (2008) 400-442. · Zbl 1159.32020 |
| [8] | A. Glutsyuk, Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety, Tr. Mat. Inst. Steklova213 (1997) 90-111; translation in Proc. Steklov Inst. Math.2(213) (1996) 83-103. · Zbl 0883.57027 |
| [9] | Ilyashenko, Y. and Yakovenko, S., Lectures on Analytic Differential Equations, , Vol. 86 (American Mathematical Society, Providence, RI, 2008), xiv+625 pp. · Zbl 1186.34001 |
| [10] | Jouanolou, J.-P., Équations de Pfaff Algébriques, , Vol. 708 (Springer, Berlin, 1979), v+255 pp. · Zbl 0477.58002 |
| [11] | Lins Neto, A., Uniformization and the Poincaré metric on the leaves of a foliation by curves, Bol. Soc. Brasil. Mat.31(3) (2000) 351-366. · Zbl 0987.37039 |
| [12] | Canille Martins, J. C. and Lins Neto, A., Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves, Trans. Amer. Math. Soc.356(7) (2004) 2963-2988. · Zbl 1069.37039 |
| [13] | Lins Neto, A. and Soares, M. G., Algebraic solutions of one-dimensional foliations, J. Differ. Geom.43(3) (1996) 652-673. · Zbl 0873.32031 |
| [14] | Loray, F. and Rebelo, J. C., Minimal, rigid foliations by curves on \(\mathbb{C} \Bbb P^n\), J. Eur. Math. Soc.5(2) (2003) 147-201. · Zbl 1021.37030 |
| [15] | Martinet, J. and Ramis, J.-P., Classification analytique des équations différentielles non-linéaires résonnantes du premier ordre [Analytic classification of first-order resonant non-linear differential equations.]Ann. Sci. École Norm. Sup.16(4) (1984) 571-621. · Zbl 0534.34011 |
| [16] | V.-A. Nguyên, Singular holomorphic foliations by curves II: Negative Lyapunov exponent, J. Geom. Anal. preprint (2018), arXiv:1812.10125v2. |
| [17] | Nguyên, V.-A., Ergodic theory for Riemann surface laminations: A survey, in Geometric Complex Analysis, , Vol. 246 (Springer, Singapore, 2018), pp. 291-327. · Zbl 1404.32057 |
| [18] | Nguyên, V.-A., Ergodic theorems for laminations and foliations: Recent results and perspectives, Acta Math. Vietnam.46(1) (2021) 9-101. · Zbl 1469.37004 |
| [19] | Palis, J. and de Melo, W., Geometric Theory of Dynamical Systems. An Introduction, Translated from the Portuguese by A. K. Manning (Springer-Verlag, New York, Berlin, 1982), xii+198 pp. · Zbl 0491.58001 |
| [20] | Verjovsky, A., A uniformization theorem for holomorphic foliations, The Lefschetz Centennial Conference, Part III (Mexico City, 1984), , Vol. 58 (American Mathematical Society, Providence, RI, 1987), pp. 233-253. · Zbl 0619.32017 |
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.
