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On the range of \({\mathbb{R}}^2\) or \({\mathbb{R}}^3\)-valued harmonic morphisms. (English) Zbl 0933.31006

Summary: The author proves that, under some general assumptions, the range of any nonconstant harmonic morphism from a simply connected open set \(U\) in \(\mathbb{R}^n\) to \(\mathbb{R}^3\), \(n>3\), cannot avoid three concurrent half-lines, which is an extension of Picard’s little theorem. To this end, he proves two results concerning the windings of Brownian motion around three concurrent half-lines in \(\mathbb{R}^3\) and the recurrence of some domains linked with the harmonic morphism.

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
60J45 Probabilistic potential theory
60J65 Brownian motion
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[1] BAIRD, P. and WOOD, J. C. 1988. Bernstein theorems for harmonic morphisms from R and S3. Math. Ann. 280 579 603. · Zbl 0621.58011
[2] BERNARD, A., CAMPBELL, E. A. and DAVIE, A. M. 1979. Brownian motion and generalized analytic and inner functions. Ann. Inst. Fourier 29 207 228. · Zbl 0386.30029
[3] DAVIS, B. 1975. Picard’s theorem and Brownian motion. Trans. Amer. Math. Soc. 213 353 362. · Zbl 0292.60126
[4] DUHEILLE, F. 1995. Sur l’image des morphismes harmoniques a valeurs dans R ou R. C.R. Acad. Sci. Paris Ser. I Math. 320 1495 1500. \' · Zbl 0836.31006
[5] DURRETT, R. 1984. Brownian Motion and Martingales in Analysis. Wadsworth, Monterey, CA. · Zbl 0554.60075
[6] FRIEDLAND, S. and HAYMAN, W. K. 1976. Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv. 51 133 161. · Zbl 0339.31003
[7] FUGLEDE, B. 1978. Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier 28 107 144. · Zbl 0339.53026
[8] GUGMUNDSSON, S. 1994. Harmonic morphisms from complex projective spaces. Geom. Dedicata 53 155 161. \" · Zbl 0826.53028
[9] HUBER, A. 1952. Uber Wachstumseigenschaften gewisser Klassen von Subharmonischen Funktionen. Comment. Math. Helv. 26 81 116. · Zbl 0049.05901
[10] KELLOGG, O. D. 1953. Foundations of Potential Theory. Dover, New York. · Zbl 0053.07301
[11] LEVY, P. 1948. Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. \' · Zbl 0034.22603
[12] MCKEAN, H. P. 1969. Stochastic Integrals. Academic Press, New York. · Zbl 0191.46603
[13] PORT, S. C. and STONE, C. J. 1978. Brownian Motion and Classical Potential Theory. Academic Press, New York. · Zbl 0413.60067
[14] UNIVERSITE CLAUDE BERNARD, LYON 1 \' 43, BOULEVARD DU 11 NOVEMBRE 1918 69622 VILLEURBANNE CEDEX FRANCE E-MAIL: duheille@jonas.univ-lyon1.fr
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