Journal of the Mathematical Society of Japan

Value distribution of leafwise holomorphic maps on complex laminations by hyperbolic Riemann surfaces

Atsushi ATSUJI

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We discuss the value distribution of Borel measurable maps which are holomorphic along leaves of complex laminations. In the case of complex lamination by hyperbolic Riemann surfaces with an ergodic harmonic measure, we have a defect relation appearing in Nevanlinna theory. It gives a bound of the number of omitted hyperplanes in general position by those maps.

Article information

J. Math. Soc. Japan, Volume 69, Number 2 (2017), 477-501.

First available in Project Euclid: 20 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32H30: Value distribution theory in higher dimensions {For function- theoretic properties, see 32A22}
Secondary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

Nevanlinna theory complex lamination value distribution theory holomorphic diffusion


ATSUJI, Atsushi. Value distribution of leafwise holomorphic maps on complex laminations by hyperbolic Riemann surfaces. J. Math. Soc. Japan 69 (2017), no. 2, 477--501. doi:10.2969/jmsj/06920477.

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  • A. Atsuji, A Casorati–Weierstrass theorem for holomorphic maps and invariant $\sigma-$fields of holomorphic diffusions, Bull. Sci. Math., 123 (1999), 371–383.
  • A. Atsuji, On the number of omitted values by a meromorphic function of finite energy and heat diffusions, Jour. Geom. Anal., 20 (2010), 1008–1025.
  • A. Atsuji, A second main theorem of Nevanlinna theory for meromorphic functions on complete Kähler manifolds, J. Math. Soc. Japan, 60 (2008), 471–493.
  • R. Bass, Probabilistic techniques in analysis, Springer, New-York, 1995.
  • B. Berndtsson and N. Sibony, The $\overline{\partial}$-equation on a positive current, Invent. Math., 147 (2002), 371–428.
  • A. Candel, The harmonic measures of Lucy Garnett, Adv. Math., 176 (2003), 187–247.
  • A. Candel, Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. (4), 26 (1993), 489–516.
  • A. Candel and L. Conlon, Foliations. II, Graduate Studies in Mathematics, 60, Amer. Math. Soc., Providence, RI, 2003.
  • A. Candel and X. Gomez-Mont, Uniformization of the leaves of a rational vector field, Ann. Inst. Fourier, 45 (1995), 1123–1133.
  • R. W. R. Darling, Convergence of martingales on a Riemannian manifold, Publ. Res. Inst. Math. Kyoto Univ., 19 (1983), 753–763.
  • T.-C. Dinh, V.-A. Nguyen and N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations, Math. Ann., 354 (2012), 331–376.
  • M. Emery, Stochastic Calculus in Manifolds, Springer, 1989.
  • R. Feres and A. Zeghib, Leafwise holomorphic functions, Proc. Amer. Math. Soc., 131 (2003), 1717–1725 (electronic).
  • J. E. Fornæ ss and N. Sibony, Unique ergodicity of harmonic currents on singular foliations of ${\bf P}^2$, Geom. Funct. Anal., 19 (2010), 1334–1377.
  • H. Fujimoto, Value Distribution Theory of the Gauss Map of Minimal Surfaces in ${\bf R}^m$, Aspects of Mathematics 21, Vieweg, 1993.
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, 1994.
  • L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51 (1983), 285–311.
  • R. K. Getoor and M. J. Sharpe, Conformal martingales, Invent. Math., 16 (1972), 271–308.
  • N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, Second edition, North-Holland Math. Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
  • A. Lins Neto, A note on projective Levi-flats and minimal sets of algebraic foliations, Ann. Inst. Fourier, 49 (1999), 1369–1385.
  • Z.-M. Ma and M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Universitext, Springer, 1992.
  • R. E. Molzon, Potential theory on complex projective space: Application to characterization of pluripolar sets and growth of analytic varieties, Illinois J. Math., 28 (1984), 103–119.
  • J. Noguchi, A note on entire pseudo-holomorphic curves and the proof of Cartan–Nochka's theorem, Kodai Math. J., 28 (2005), 336–346.
  • J. Noguchi and T. Ochiai, Geometric Function Theory in Several Complex Variables, Amer. Math. Soc., Providence, RI, 1997.
  • D. Revuz and M. Yor, Continuous martingales and Brownian motion, Springer, Berlin, 1991.
  • M. Ru, Nevanlinna Theory and It's Relation to Diophantine Approximation, World Scientific Pub., 2001.
  • L. Schwartz, Semi-martingales sur des variétés analytiques complexes, Lecture notes in Math. 780, Springer, 1980.
  • Y.-T. Siu, Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension $\ge 3$, Ann. Math., 151 (2000), 1217–1243.
  • D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225–255.
  • S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure and App. Math., 28 (1975), 201–228.