A general Hopf lemma and proper holomorphic mappings between convex domains in 𝐶ⁿ

Mercer, Peter R.

doi:10.1090/S0002-9939-1993-1152283-7

A general Hopf lemma and proper holomorphic mappings between convex domains in $\textbf {C}^ n$
HTML articles powered by AMS MathViewer

by Peter R. Mercer

Proc. Amer. Math. Soc. 119 (1993), 573-578

DOI: https://doi.org/10.1090/S0002-9939-1993-1152283-7

PDF | Request permission

Abstract:

We use a general version of the well-known Hopf lemma to study boundary regularity of proper holomorphic mappings between some bounded convex domains in ${\mathbb {C}^n}$ which carry no boundary regularity assumption.

References

François Berteloot, Hölder continuity of proper holomorphic mappings, Studia Math. 100 (1991), no. 3, 229–235. MR 1133387, DOI 10.4064/sm-100-3-229-235
Klas Diederich and John E. Fornæss, Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary, Ann. of Math. (2) 110 (1979), no. 3, 575–592. MR 554386, DOI 10.2307/1971240
John Erik Fornæss and Berit Stensønes, Lectures on counterexamples in several complex variables, Mathematical Notes, vol. 33, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1987. MR 895821
Franc Forstnerič, Proper holomorphic mappings: a survey, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 297–363. MR 1207867
Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $C^{n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219–240. MR 372252, DOI 10.1090/S0002-9947-1975-0372252-8
Ian Graham, Distortion theorems for holomorphic maps between convex domains in $\textbf {C}^n$, Complex Variables Theory Appl. 15 (1990), no. 1, 37–42. MR 1055937, DOI 10.1080/17476939008814431
Ian Graham, Sharp constants for the Koebe theorem and for estimates of intrinsic metrics on convex domains, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 233–238. MR 1128547, DOI 10.1090/pspum/052.2/1128547
G. M. Henkin, An analytic polyhedron is not holomorphically equivalent to a strictly pseudoconvex domain, Dokl. Akad. Nauk SSSR 210 (1973), 1026–1029 (Russian). MR 0328125
Shoshichi Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), no. 3, 357–416. MR 414940, DOI 10.1090/S0002-9904-1976-14018-9
Steven G. Krantz, Optimal Lipschitz and $L^{p}$ regularity for the equation $\overline \partial u=f$ on stongly pseudo-convex domains, Math. Ann. 219 (1976), no. 3, 233–260. MR 397020, DOI 10.1007/BF01354286
Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
Pierre Lelong and Lawrence Gruman, Entire functions of several complex variables, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 282, Springer-Verlag, Berlin, 1986. MR 837659, DOI 10.1007/978-3-642-70344-7
László Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), no. 4, 427–474 (French, with English summary). MR 660145, DOI 10.24033/bsmf.1948

Complex geodesics and iterates of holomorphic maps on convex domains in

Keith Miller, Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl. (4) 76 (1967), 93–105 (English, with Italian summary). MR 221087, DOI 10.1007/BF02412230
Keith Miller, Extremal barriers on cones with Phragmén-Lindelöf theorems and other applications, Ann. Mat. Pura Appl. (4) 90 (1971), 297–329. MR 316884, DOI 10.1007/BF02415053
J. K. Oddson, On the boundary point principle for elliptic equations in the plane, Bull. Amer. Math. Soc. 74 (1968), 666–670. MR 227611, DOI 10.1090/S0002-9904-1968-11985-8

On proper holomorphic mappings of strictly pseudoconvex domains

15

R. Michael Range, On the topological extension to the boundary of biholomorphic maps in $C^{n}$, Trans. Amer. Math. Soc. 216 (1976), 203–216. MR 387665, DOI 10.1090/S0002-9947-1976-0387665-9
R. Michael Range, The Carathéodory metric and holomorphic maps on a class of weakly pseudoconvex domains, Pacific J. Math. 78 (1978), no. 1, 173–189. MR 513293, DOI 10.2140/pjm.1978.78.173
H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin, 1971, pp. 125–137. MR 0304694
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6