Abstract
LetX(-ϱB m×C n be a compact set over the unit sphere ϱB m such that for eachz∈ϱB m the fiberX z ={ω∈C n;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC n. The polynomial hull\(\hat X\) ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z 0,w 0)∈Int\(\hat X\) there exists a smooth up to the boundary analytic discF:Δ→B m×C n with the boundary inX such thatF(0)=(z 0,w 0).
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References
Alexander, H., Polynomial hulls of graphs,Pacific J. Math. 147 (1991), 201–212.
Alexander, H. andWermer, J., Polynomial hulls with convex fibers,Math. Ann. 266 (1981), 243–257.
Bedford, E., Survey of pluri-potential theory, inSeveral Complex Variables: Proceedings of the Mittag-Leffler Institute, 1987–1988 (Fornæss, J. E., ed.), Mathematical Notes38, pp. 48–97, Princeton Univ. Press, Princeton, N. J., 1993.
Bedford, E. andKalka, M., Foliations and complex Monge-Ampère quations,Comm. Pure Appl. Math. 30 (1977), 543–571.
Bedford, E. andTaylor, 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. Monographs6, 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, inSeveral 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 inC 2 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 inC 2,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 andH ∞ 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.
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This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.
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Černe, M. Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration. Ark. Mat. 40, 27–45 (2002). https://doi.org/10.1007/BF02384500
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DOI: https://doi.org/10.1007/BF02384500