## Arkiv för Matematik

### Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

Miran Černe

#### Abstract

LetX(-ϱBm×Cn be a compact set over the unit sphere ϱBm such that for each z∈ϱBm the fiber Xz={ω∈Cn;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain in Cn. The polynomial hull $\hat X$ of X is described in terms of the Perron-Bremermann function for the homogeneous defining function of X. Moreover, for each point (z0, w0)∈Int $\hat X$ there exists a smooth up to the boundary analytic disc F:Δ→Bm×Cn with the boundary in X such that F(0)=(z0, w0).

#### Note

This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.

#### Article information

Source
Ark. Mat., Volume 40, Number 1 (2002), 27-45.

Dates
Revised: 22 June 2001
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485898751

Digital Object Identifier
doi:10.1007/BF02384500

Mathematical Reviews number (MathSciNet)
MR1948884

Zentralblatt MATH identifier
1039.32014

Rights

#### Citation

Černe, Miran. Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration. Ark. Mat. 40 (2002), no. 1, 27--45. doi:10.1007/BF02384500. https://projecteuclid.org/euclid.afm/1485898751

#### References

• Alexander, H., Polynomial hulls of graphs, Pacific J. Math. 147 (1991), 201–212.
• Alexander, H. and Wermer, J., Polynomial hulls with convex fibers, Math. Ann. 266 (1981), 243–257.
• Bedford, E., Survey of pluri-potential theory, in Several Complex Variables: Proceedings of the Mittag-Leffler Institute, 1987–1988 (Fornæss, J. E., ed.), Mathematical Notes 38, pp. 48–97, Princeton Univ. Press, Princeton, N. J., 1993.
• Bedford, E. and Kalka, M., Foliations and complex Monge-Ampère quations, Comm. Pure Appl. Math. 30 (1977), 543–571.
• Bedford, E. and Taylor, B. A., The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), 129–134.
• Černe, M., Stationary discs of fibrations over the circle, Internat. J. Math. 6 (1995), 805–823.
• Černe, M., Analytic varieties with boundaries in totally real tori, Michigan Math. J. 45 (1998), 243–256.
• Černe, M., Analytic discs in the polynomial hull of a disc fibration over the sphere, Bull. Austral. Math. Soc., 62, (2000), 403–406.
• Forstnerič, F., Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J. 37 (1988), 869–889.
• Gamelin, T. W., Uniform Algebras and Jensen Measures, London Math. Soc. Lecture Notes Ser. 32, Cambridge Univ. Press, Cambridge-New York, 1978.
• Garnett, J. B., Bounded Analytic Functions, Academic Press, Orlando, Fla. 1981.
• Klimek, M., Pluripotential Theory, London Math. Soc. Monographs 6, Oxford Univ. Press, Oxford, 1991.
• Lelong, P., Fonction de Green pluricomplexe et lemmes de Schwarz dans les espaces de Banach, J. Math. Pures Appl. 68 (1989), 319–347.
• Poletsky E. A., Plurisubharmonic functions as solutions of variational problems, in Several Complex Variables and Complex Geometry (Santa Cruz, Calif., 1989) (Bedford, E., D'Angelo, J. P., Greene, R. E. and Krantz, S. G., eds.), Proc. Symp. Pure Math. 52, Part 1, pp. 163–171, Amer. Math. Soc., Providence, R. I., 1991.
• Poletsky, E. A., Holomorphic currents, Indiana Univ. Math. J. 42 (1993), 85–144.
• Slodkowski, Z., Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), 255–260.
• Slodkowski, Z., Polynomial hulls in C2 and quasicircles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1989), 367–391.
• Slodkowski, Z., Polynomial hulls with convex fibers and complex geodesics, J. Funct. Anal. 94, (1990), 156–176.
• Walsh, J. B., Continuity of envelopes of plurisubharmonic functions, J. Math. Mech. 18 (1968), 143–148.
• Whittlesey, M. A., Polynomial hulls with disk fibers over the ball in C2, Michigan Math. J. 44 (1997), 475–494.
• Whittlesey, M. A., Riemann surfaces in fibered polynomial hulls, Ark. Mat. 37 (1999), 409–423.
• Whittlesey, M. A., Polynomial hulls and H control for a hypoconvex constraint, Math. Ann. 317 (2000), 677–701.
• Whittlesey, M. A., Polynomial hulls, an optimization problem and the Kobayashi metric in a hypoconvex domain, Preprint, 1999.