## Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary

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- by Daniel M. Burns and Steven G. Krantz
- J. Amer. Math. Soc.
**7**(1994), 661-676 - DOI: https://doi.org/10.1090/S0894-0347-1994-1242454-2
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## Abstract:

A rigidity theorem for holomorphic mappings, in the nature of the uniqueness statement of the classical one-variable Schwarz lemma, is proved at the boundary of a strongly pseudoconvex domain. The result reduces to an interesting, and apparently new, result even in one complex dimension. The theorem has a variety of geometric and analytic interpretations.## References

- Lars V. Ahlfors,
*Conformal invariants: topics in geometric function theory*, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR**0357743** - H. Alexander,
*Holomorphic mappings from the ball and polydisc*, Math. Ann.**209**(1974), 249–256. MR**352531**, DOI 10.1007/BF01351851 - L. A. Aĭzenberg,
*Multidimensional analogues of the Carleman formula with integration over boundary sets of maximal dimension*, Multidimensional complex analysis (Russian), Akad. Nauk SSSR Sibirsk. Otdel., Inst. Fiz., Krasnoyarsk, 1985, pp. 12–22, 272 (Russian). MR**899332** - Elisabetta Barletta and Eric Bedford,
*Existence of proper mappings from domains in $\textbf {C}^2$*, Indiana Univ. Math. J.**39**(1990), no. 2, 315–338. MR**1089041**, DOI 10.1512/iumj.1990.39.39018
D. Burns, - D. Burns Jr., S. Shnider, and R. O. Wells Jr.,
*Deformations of strictly pseudoconvex domains*, Invent. Math.**46**(1978), no. 3, 237–253. MR**481119**, DOI 10.1007/BF01390277 - S. S. Chern and J. K. Moser,
*Real hypersurfaces in complex manifolds*, Acta Math.**133**(1974), 219–271. MR**425155**, DOI 10.1007/BF02392146 - Charles Fefferman,
*The Bergman kernel and biholomorphic mappings of pseudoconvex domains*, Invent. Math.**26**(1974), 1–65. MR**350069**, DOI 10.1007/BF01406845 - John Erik Fornaess,
*Embedding strictly pseudoconvex domains in convex domains*, Amer. J. Math.**98**(1976), no. 2, 529–569. MR**422683**, DOI 10.2307/2373900
E. Gavosto, Thesis, Washington University, 1990.
- 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 - R. E. Greene and Steven G. Krantz,
*Stability properties of the Bergman kernel and curvature properties of bounded domains*, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 179–198. MR**627758** - Robert E. Greene and Steven G. Krantz,
*Deformation of complex structures, estimates for the $\bar \partial$ equation, and stability of the Bergman kernel*, Adv. in Math.**43**(1982), no. 1, 1–86. MR**644667**, DOI 10.1016/0001-8708(82)90028-7
—, - Xiao Jun Huang,
*Some applications of Bell’s theorem to weakly pseudoconvex domains*, Pacific J. Math.**158**(1993), no. 2, 305–315. MR**1206440**
—, - Yitzhak Katznelson,
*An introduction to harmonic analysis*, Second corrected edition, Dover Publications, Inc., New York, 1976. MR**0422992** - Shoshichi Kobayashi,
*Hyperbolic manifolds and holomorphic mappings*, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR**0277770** - Steven G. Krantz,
*Function theory of several complex variables*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR**635928**
—, - J. J. Kohn and Hugo Rossi,
*On the extension of holomorphic functions from the boundary of a complex manifold*, Ann. of Math. (2)**81**(1965), 451–472. MR**177135**, DOI 10.2307/1970624 - A. G. Vitushkin, V. V. Ezhov, and N. G. Kruzhilin,
*Extension of local mappings of pseudoconvex surfaces*, Dokl. Akad. Nauk SSSR**270**(1983), no. 2, 271–274 (Russian). MR**712528** - 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** - L. Lempert,
*Holomorphic retracts and intrinsic metrics in convex domains*, Anal. Math.**8**(1982), no. 4, 257–261 (English, with Russian summary). MR**690838**, DOI 10.1007/BF02201775
—, - S. I. Pinčuk,
*The analytic continuation of holomorphic mappings*, Mat. Sb. (N.S.)**98(140)**(1975), no. 3(11), 416–435, 495–496 (Russian). MR**0393562** - Walter Rudin,
*Holomorphic maps that extend to automorphisms of a ball*, Proc. Amer. Math. Soc.**81**(1981), no. 3, 429–432. MR**597656**, DOI 10.1090/S0002-9939-1981-0597656-8
—, - A. G. Vitushkin,
*Real-analytic hypersurfaces of complex manifolds*, Uspekhi Mat. Nauk**40**(1985), no. 2(242), 3–31, 237 (Russian). MR**786085** - Warren R. Wogen,
*Composition operators acting on spaces of holomorphic functions on domains in $\textbf {C}^n$*, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 361–366. MR**1077457**, DOI 10.1090/pspum/051.2/1077457 - H. Wu,
*Normal families of holomorphic mappings*, Acta Math.**119**(1967), 193–233. MR**224869**, DOI 10.1007/BF02392083 - Shing Tung Yau,
*A general Schwarz lemma for Kähler manifolds*, Amer. J. Math.**100**(1978), no. 1, 197–203. MR**486659**, DOI 10.2307/2373880

*A multi-valued Hartogs theorem and developing maps*, preprint.

*Methods for studying the automorphism groups of weakly pseudoconvex domains*, Proc. Internat. Conf. on Complex Geometry (Cetraro, Italy), Mediterranean Press, Calabria, 1991.

*A boundary rigidity problem of holomorphic mappings on weakly pseudoconvex domains*, preprint. —,

*Preservation principle of extremal mappings and its applications*, Illinois J. Math. (to appear).

*A new compactness principle in complex analysis*, Univ. Autonoma de Madrid, 1987.

*Intrinsic distances and holomorphic retracts*, Complex Analysis and Applications, Sofia, 1984.

*Function theory of the unit ball in*${\mathbb {C}^n}$, Springer-Verlag, Berlin and New York, 1980. N. Sibony, Unpublished notes. J. Velling, Thesis, Stanford, 1985.

## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**7**(1994), 661-676 - MSC: Primary 32H02; Secondary 32H15
- DOI: https://doi.org/10.1090/S0894-0347-1994-1242454-2
- MathSciNet review: 1242454