## Generalized Ahlfors functions

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- by Miran Černe and Manuel Flores PDF
- Trans. Amer. Math. Soc.
**359**(2007), 671-686 Request permission

## 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 ${\mathcal F}$.## References

- Lars V. Ahlfors,
*Bounded analytic functions*, Duke Math. J.**14**(1947), 1–11. MR**21108** - Lars V. Ahlfors,
*Open Riemann surfaces and extremal problems on compact subregions*, Comment. Math. Helv.**24**(1950), 100–134. MR**36318**, DOI 10.1007/BF02567028 - H.Alexander and J.Wermer,
*Polynomial hulls with convex fibers*, Math. Ann.**266**(1981), 243–257. - David E. Barrett,
*Failure of averaging on multiply connected domains*, Ann. Inst. Fourier (Grenoble)**40**(1990), no. 2, 357–370 (English, with French summary). MR**1070831** - H. Begehr and M. A. Efendiev,
*On the asymptotics of meromorphic solutions for nonlinear Riemann-Hilbert problems*, Math. Proc. Cambridge Philos. Soc.**127**(1999), no. 1, 159–172. MR**1692479**, DOI 10.1017/S0305004199003539 - Steven R. Bell,
*Finitely generated function fields and complexity in potential theory in the plane*, Duke Math. J.**98**(1999), no. 1, 187–207. MR**1687563**, DOI 10.1215/S0012-7094-99-09805-8 - Steven 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**, DOI 10.1007/BF02791140 - Steven R. Bell,
*A Riemann surface attached to domains in the plane and complexity in potential theory*, Houston J. Math.**26**(2000), no. 2, 277–297. MR**1814239** - Steven R. Bell,
*Complexity in complex analysis*, Adv. Math.**172**(2002), no. 1, 15–52. MR**1943900**, DOI 10.1006/aima.2002.2076 - Errett Bishop,
*Subalgebras of functions on a Riemann surface*, Pacific J. Math.**8**(1958), 29–50. MR**96818** - Young-Bok Chung,
*Higher order extremal problem and proper holomorphic mapping*, Houston J. Math.**27**(2001), no. 3, 707–718. MR**1864806** - Miran Černe,
*Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces*, Amer. J. Math.**126**(2004), no. 1, 65–87. MR**2033564** - Miran Černe and Franc Forstnerič,
*Embedding some bordered Riemann surfaces in the affine plane*, Math. Res. Lett.**9**(2002), no. 5-6, 683–696. MR**1906070**, DOI 10.4310/MRL.2002.v9.n5.a10 - Miran Černe and Josip Globevnik,
*On holomorphic embedding of planar domains into $\textbf {C}^2$*, J. Anal. Math.**81**(2000), 269–282. MR**1785284**, DOI 10.1007/BF02788992 - E. M. Chirka,
*Regularity of the boundaries of analytic sets*, Mat. Sb. (N.S.)**117(159)**(1982), no. 3, 291–336, 431 (Russian). MR**648411** - Jeffrey Diller,
*Failure of weak holomorphic averaging on multiple connected domains*, Math. Z.**217**(1994), no. 2, 167–177. MR**1296392**, DOI 10.1007/BF02571940 - M. A. Efendiev and W. L. Wendland,
*Nonlinear Riemann-Hilbert problems for multiply connected domains*, Nonlinear Anal.**27**(1996), no. 1, 37–58. MR**1390711**, DOI 10.1016/0362-546X(94)00354-K - M. A. Efendiev and W. L. Wendland,
*Nonlinear Riemann-Hilbert problems without transversality*, Math. Nachr.**183**(1997), 73–89. MR**1434976**, DOI 10.1002/mana.19971830106 - 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), no. 1, 111–124. MR**1846981**, DOI 10.12775/TMNA.2001.007 - Franc Forstnerič,
*Polynomial hulls of sets fibered over the circle*, Indiana Univ. Math. J.**37**(1988), no. 4, 869–889. MR**982834**, DOI 10.1512/iumj.1988.37.37042 - M.Gromov,
*Pseudo-holomorphic curves in symplectic manifolds*, Invent. Math.**81**(1985), 307–347. - C. Denson Hill and Geraldine Taiani,
*Families of analytic discs in $\textbf {C}^{n}$ with boundaries on a prescribed CR submanifold*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**5**(1978), no. 2, 327–380. MR**501906** - Laura Ketchum Kodama,
*Boundary measures of analytic differentials and uniform approximation on a Riemann surface*, Pacific J. Math.**15**(1965), 1261–1277. MR**190327** - Walter Koppelman,
*The Riemann-Hilbert problem for finite Riemannian surfaces*, Comm. Pure Appl. Math.**12**(1959), 13–35. MR**146394**, DOI 10.1002/cpa.3160120103 - F. G. Maksudov and M. A. Èfendiev,
*The nonlinear Hilbert problem for a doubly connected domain*, Dokl. Akad. Nauk SSSR**290**(1986), no. 4, 789–791 (Russian). MR**863355** - Pierre Pansu,
*Compactness*, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 233–249. MR**1274932**, DOI 10.1007/978-3-0348-8508-9_{9} - E. A. Poletskiĭ,
*The Euler-Lagrange equations for extremal holomorphic mappings of the unit disk*, Michigan Math. J.**30**(1983), no. 3, 317–333. MR**725784**, DOI 10.1307/mmj/1029002908 - Menahem Schiffer and Donald C. Spencer,
*Functionals of finite Riemann surfaces*, Princeton University Press, Princeton, N. J., 1954. MR**0065652** - Zbigniew Slodkowski,
*Polynomial hulls in $\textbf {C}^2$ and quasicircles*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**16**(1989), no. 3, 367–391 (1990). MR**1050332** - A. I. Šnirel′man,
*The degree of a quasiruled mapping, and the nonlinear Hilbert problem*, Mat. Sb. (N.S.)**89(131)**(1972), 366–389, 533 (Russian). MR**0326521** - Elias Wegert,
*Nonlinear boundary value problems for holomorphic functions and singular integral equations*, Mathematical Research, vol. 65, Akademie-Verlag, Berlin, 1992 (English, with English and German summaries). MR**1206907** - Rugang Ye,
*Gromov’s compactness theorem for pseudo holomorphic curves*, Trans. Amer. Math. Soc.**342**(1994), no. 2, 671–694. MR**1176088**, DOI 10.1090/S0002-9947-1994-1176088-1

## Additional Information

**Miran Černe**- Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, 1 111 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
- 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 - © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2255192