Abstract
We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.
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References
Bernal-González, L.: Universal entire functions for affine endomorphisms of \(\mathbb {C}^{N}\). (English summary). J. Math. Anal. Appl. 305(2), 690–697 (2005)
Berndtsson, B., Sibony, N: The \(\overline \partial \)-equation on a positive current. Invent. Math. 147(2), 371–428 (2002)
Birkhoff, G.D.: Démonstration d’un théorème élémentaire sur les fonctions entières. C.R. Acad. Sci. Paris 189, 473–475 (1929)
Bogomolov, F.: Complex manifolds and algebraic foliations. Publ. RIMS, vol. 1084. Kyoto. Unpublished (1996)
Brunella, M.: Inexistence of invariant measures for generic rational differential equations in the complex domain. Bol. Soc. Mat. Mexicana (3) 12(1), 43–49 (2006)
Buhovsky, L., Glucksam, A., Logunov, A., Sodin, M.: Translation-invariant probability measures on entire functions. Preprint (2017), 38pp. arXiv:1703.08101
Burns, D., Sibony, N.: Limit currents and value distribution of holomorphic maps. Ann. Inst. Fourier (Grenoble) 62(1), 145–176 (2012)
Camacho, C., Lins Neto, A., Sad, P.: Minimal sets of foliations on complex projective spaces. Inst. Hautes Études Sci. Publ. Math. 68, 187–203 (1988)
Cerveau, D., Lins Neto, A.: Irreducible components of the space of foliations of degree two in \(\mathbb {P}^{n}\), n ≥ 3. Ann. Math. 143, 577–612 (1996)
Demailly, J. -P., Gaussier, H.: Algebraic embeddings of smooth almost complex structures. J. Eur. Math. Soc. (JEMS) 19(11), 3391–3419 (2017)
Dinh, T.-C., Nguyen, V.-A., Sibony, N.: Heat equation and ergodic theorems for Riemann surface laminations. Math. Ann. 354(1), 331–376 (2012)
Dinh, T.-C., Nguyen, V.-A., Sibony, N.: Entropy for Hyperbolic Riemann Surface Laminations I. Frontiers in Complex Dynamics, Princeton Math Ser., vol. 51, pp 569–592. Princeton Univ. Press, Princeton (2014)
Dinh, T.-C., Nguyen, V.-A., Sibony, N.: Unique Ergodicity for foliations on compact Kähler surfaces. Preprint (2018)
Dinh, T.-C., Sibony, N.: Unique ergodicity for foliations in \(\mathbb {P}^{2}\) with an invariant curve. Invent. Math. 211(1), 1–38 (2018)
Fornæss, J.-E., Sibony, N.: Harmonic currents of finite energy and laminations. Geom. Funct. Anal. 15(5), 962–1003 (2005)
Fornæss, J.-E., Sibony, N.: Riemann surface laminations with singularities. J. Geom. Anal. 18(2), 400–442 (2008)
Fornæss, J.-E., Sibony, N., Wold, E.-F.: Examples of minimal laminations and associated currents. Math. Z. 269(1), 495–520 (2011)
Furstenberg, H.: Strict ergodicity and transformation of the torus. Am. J. Math. 83, 573–601 (1961)
Garnett, L.: Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51, 285–311 (1983)
Ilyashenko, Yu: Some open problems in real and complex dynamical systems. Nonlinearity 21(7), T101–T107 (2008)
Jouanolou, J.P.: Feuilles compactes des feuilletages algébriques. (French). Math. Ann. 241(1), 69–72 (1979)
Lins Neto, A.: Uniformization and the Poincaré metric on the leaves of a foliation by curves. Bol. Soc. Brasil. Mat. (N.S.) 31(3), 351–366 (2000)
Lins Neto, A., Soares, M.: Algebraic solutions of one dimensional foliations. J. Diff. Geom. 43, 652–673 (1996)
Nguyên, V.-A.: Oseledec multiplicative ergodic theorem for laminations. Mem. Am. Math. Soc. 246, 1164 (2017)
Nguyên, V.-A.: Singular holomorphic foliations by curves I: integrability of holonomy cocycle in dimension 2. Invent. Math. 212(2), 531–618 (2018)
Nguyên, V.-A.: Ergodic theory for Riemann surface laminations: a survey. In: Byun, J., Cho, H., Kim, S., Lee, K.H., Park, J.D. (eds.) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol. 246, pp 291–327. Springer, Singapore (2018)
Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes I-V. J. Math. Kyoto Univ. 8, 9, 10, 13, 15 (1968-1975)
Shcherbakov, A.A., Rosales-González, E., Ortiz-Bobadilla, L.: Countable set of limit cycles for the equation dw/dz = Pn(z,w)/Qn(z,w). J. Dynam. Control Syst. 4(4), 539–581 (1998)
Sibony, N.: Pfaff systems, currents and hulls. Math. Z. 285(3–4), 1107–1123 (2017)
Sibony, N., Wold, E.F.: Topology and complex structure of leaves of foliations by Riemann surfaces. J. Geom. Anal (2017)
Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)
Weiss, B.: Measurable entire functions. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin. Ann. Numer. Math. 4(1–4), 599–605 (1997)
Yamaguchi, H.: Parabolicité d’une fonction entière. J. Math. Kyoto Univ. 16, 71–92 (1976)
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The first author was supported by the NUS grants C-146-000-047-001 and AcRF Tier 1 R-146-000-248-114.
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Dinh, TC., Sibony, N. Some Open Problems on Holomorphic Foliation Theory. Acta Math Vietnam 45, 103–112 (2020). https://doi.org/10.1007/s40306-018-00323-0
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DOI: https://doi.org/10.1007/s40306-018-00323-0