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Generalized Ahlfors functions


Authors: Miran Cerne and Manuel Flores
Journal: Trans. Amer. Math. Soc. 359 (2007), 671-686
MSC (2000): Primary 35Q15; Secondary 32E99, 30E25
DOI: https://doi.org/10.1090/S0002-9947-06-03906-7
Published electronically: July 20, 2006
MathSciNet review: 2255192
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Sigma$ be a bordered Riemann surface with genus $ g$ and $ m$ boundary components. Let $ \lbrace\gamma_{z}\rbrace_{z\in\partial\Sigma}$ be a smooth family of smooth Jordan curves in $ \mathbb{C}$ which all contain the point 0 in their interior. Let $ p\in\Sigma$ and let $ {\mathcal F}$ be the family of all bounded holomorphic functions $ f$ on $ \Sigma$ such that $ f(p)\ge 0$ and $ f(z)\in \widehat{\gamma_z}$ for almost every $ z\in\partial\Sigma$. Then there exists a smooth up to the boundary holomorphic function $ f_0\in {\mathcal F}$ with at most $ 2g+m-1$ zeros on $ \Sigma$ so that $ f_0(z)\in\gamma_z$ for every $ z\in\partial\Sigma$ and such that $ f_0(p)\ge f(p)$ for every $ f\in {\mathcal F}$. If, in addition, all the curves $ \lbrace\gamma_z\rbrace_{z\in\partial\Sigma}$ are strictly convex, then $ f_0$ is unique among all the functions from the family $ {\mathcal F}$.


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  • 1. L.Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1-11. MR 0021108 (9:24a)
  • 2. L.Ahlfors, Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv. 24 (1950), 100-134. MR 0036318 (12:90b)
  • 3. H.Alexander and J.Wermer, Polynomial hulls with convex fibers, Math. Ann. 266 (1981), 243-257.
  • 4. D.E.Barrett, Failure of averaging on multiply connected domains, Ann. Inst. Fourier 40 (1990), 357-370. MR 1070831 (92b:30050)
  • 5. H.Begehr and M.A.Efendiev, On the asymptotics of meromorphic solutions for nonlinear Riemann-Hilbert problems, Math. Proc. Cambridge Philos. Soc. 127 (1999), 159-172. MR 1692479 (2000c:30076)
  • 6. S.R.Bell, Finitely generated function fields and complexity in potential theory in the plane, Duke Math. J. 98 (1999), 187-207. MR 1687563 (2000i:30015)
  • 7. S.R.Bell, Ahlfors maps, the double of a domain, and complexity in potential theory and conformal mapping, J. Anal. Math. 78 (1999), 329-344. MR 1714417 (2000m:30012)
  • 8. S.R.Bell, A Riemann surface attached to domains in the plane and complexity in potential theory, Houston J. Math. 26 (2000), 277-297. MR 1814239 (2001m:30009)
  • 9. S.R.Bell, Complexity in complex analysis, Adv. Math. 172 (2002), 15-52. MR 1943900 (2003m:30016)
  • 10. E.Bishop, Subalgebras of functions on a Riemann surface, Pacific J. Math. 8 (1958), 29-50. MR 0096818 (20:3300)
  • 11. Y.-B.Chung, Higher order extremal problem and proper holomorphic mapping, Houston J. Math. 27 (2001), 707-718. MR 1864806 (2002j:30031)
  • 12. M.Cerne, Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces, Amer. J. Math. 126 (2004), 65-85. MR 2033564 (2004k:30092)
  • 13. M.Cerne and F.Forstneric, Embedding some bordered Riemann surfaces in the affine plane, Math. Res. Lett. 9 (2002), 683-696. MR 1906070 (2003j:32021)
  • 14. M.Cerne and J.Globevnik, On holomorphic embedding of planar domains into $ \mathbb{C}^2$, J. Anal. Math. 81 (2000), 269-282. MR 1785284 (2001g:32047)
  • 15. E.M.Cirka, Regularity of boundaries of analytic sets (Russian) Math. Sb. (NS) 117 (1982), 291-334. MR 0648411 (83f:32009)
  • 16. J.Diller Failure of weak holomorphic averaging on multiple connected domains, Math. Z. 217 (1994), 167-177. MR 1296392 (95m:30034)
  • 17. M.A.Efendiev and W.L.Wendland, Nonlinear Riemann-Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), 37-58. MR 1390711 (97h:30057)
  • 18. M.A.Efendiev and W.L.Wendland, Nonlinear Riemann-Hilbert problems without transversality, Math. Nachr. 183 (1997), 73-89. MR 1434976 (98b:30038)
  • 19. M.A.Efendiev and W.L.Wendland, Nonlinear Riemann-Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), 111-124. MR 1846981 (2002d:30045)
  • 20. F.Forstneric, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J. 37 (1988), 869-889. MR 0982834 (90g:32018)
  • 21. M.Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 81 (1985), 307-347.
  • 22. C.D.Hill and G.Taiani, Families of analytic discs in $ \mathbb{C}^n$ with boundaries on a prescribed CR submanifold, Ann. Scuola. Norm. Sup. Pisa 5 (1978), 327-380. MR 0501906 (80c:32023)
  • 23. L.K.Kodama, Boundary measures of analytic differentials and uniform approximation on a Riemann surface, Pacific J. Math. 15 (1965), 1261-1277. MR 0190327 (32:7740)
  • 24. W.Koppelman, The Riemann-Hilbert problem for finite Riemann surfaces, Comm. Pure Appl. Math. 12 (1959), 13-35. MR 0146394 (26:3916)
  • 25. F.G.Maksudov, M.A.Efendiev, The nonlinear Hilbert problem for a doubly connected domain (Russian), Dokl. Akad. Nauk SSSR 290 (1986), 789-791. MR 0863355 (88j:30088)
  • 26. P.Pansu, Compactness, in M.Audin and J.Lafontaine, Editors, Holomorphic curves in symplectic geometry, Progress in Mathematics 117, Birkhäuser (1994), 233-249. MR 1274932
  • 27. E.A.Poletski{\u{\i\/}}\kern.15em, The Euler-Lagrange equations for extremal holomorphic mappings of the unit disk, Michigan Math. J. 30 (1983), 317-333. MR 0725784 (85b:32038)
  • 28. M.Schiffer and D.C.Spencer, Functionals on finite Riemann surfaces, Princeton Univ. Press., Princeton, 1954. MR 0065652 (16:461g)
  • 29. Z.Slodkowski, Polynomial hulls in $ \mathbb{C}^2$ and quasicircles, Ann. Scuola Norm. Sup. Pisa 16 (1989), 367-391. MR 1050332 (91m:32016)
  • 30. A.I.Šnirelman, The degree of a quasiruled mapping and a nonlinear Hilbert problem (Russian) Mat. Sb. 18 (1972), 373-396. MR 0326521 (48:4865)
  • 31. E.Wegert, Nonlinear boundary value problems for holomorphic functions and singular integral equations, Mathematical Research 65, Akademie-Verlag, Berlin, 1992. MR 1206907 (94b:30049)
  • 32. R.Ye, Gromov's compactness theorem for pseudo holomorphic curves, Trans. Amer. Math. Soc. 342 (1994), 671-694. MR 1176088 (94f:58030)

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Additional Information

Miran Cerne
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia
Email: miran.cerne@fmf.uni-lj.si

Manuel Flores
Affiliation: Department of Mathematics, University of La Laguna, 38771 La Laguna, Tenerife, Spain
Email: mflores@ull.es

DOI: https://doi.org/10.1090/S0002-9947-06-03906-7
Keywords: Bordered Riemann surface, Ahlfors function, Riemann-Hilbert problem
Received by editor(s): June 21, 2004
Received by editor(s) in revised form: November 22, 2004
Published electronically: July 20, 2006
Additional Notes: The first author was supported in part by a grant “Analiza in geometrija” P1-0291 from the Ministry of Education, Science and Sport of the Republic of Slovenia. Part of this work was done while the author was visiting the University of La Laguna, Tenerife, Spain. He wishes to thank the faculty of the Analysis Department for their hospitality and support.
The second author was supported in part by grants from FEDER y Ministerio de Ciencia y Tecnologia number BFM2001-3894 and Consejeria de Educacion Cultura y Deportes del Gobierno de Canarias, PI 2003/068
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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