THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS $B_{p}^{n}$ OF $\mathbb{C}^{n}$p OF Cn

doi:10.1007/s10473-024-0304-y

Abstract

Abstract: By introducing the Carathéodory metric, we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit $p$-ball $B_{p}^{n}$ of $\mathbb{C}^n$. Furthermore, the boundary rigidity theorem for holomorphic self-mappings defined on $B_{p}^{n}$ is obtained. These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for $p=2$, and the unit polydisk for $p=\infty$, respectively.

Key words: Schwarz lemma, Carathéodory metric;, rigidity

CLC Number:

32H02

Jianfei WANG, Yanhui ZHANG. THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS $B_{p}^{n}$ OF $\mathbb{C}^{n}$p OF Cn[J].Acta mathematica scientia,Series B, 2024, 44(3): 839-850.

Trendmd

References

[1] Avkhadiev F, Wirths K.Schwarz-Pick Type Inequalities. Basel: Birkh$\ddot{\rm a}$user, 2009
[2] Bonk M. On Bloch's constant. Proc Amer Math Soc, 1990, 110(4): 889-894
[3] Bracci F, Trapani S. Notes on pluripotential theory. Rend Mat, Serie VII, 2007, 2007: 197-264
[4] Burns D, Krantz S. Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J Amer Math Soc, 1994, 7: 661-676
[5] Chen H, Nie X. Schwarz lemma: the case of equality and an extension. J Geom Anal, 2021, https://doi.org/10.1007/s12220-021-00771-5
[6] Garnett J. Bounded Analytic Functions.New York: Academic Press, 1981
[7] Osserman R. A sharp Schwarz inequality on the boundary. Proc Amer Math Soc, 2000, 128: 3513-3517
[8] Gong S.Convex and Starlike Mappings in Several Complex Variables. Beijing: Science Press, 1998
[9] Graham I, Hamada H, Kohr G. A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings. J Anal Math, 2020, 140: 31-53
[10] Hamada H. A Schwarz lemma at the boundary using the Julia-Wolff-Carathéodory type condition on finite dimensional irreducible bounded symmetric domains. J Math Anal Appl, 2018, 465(1): 196-210
[11] Hamada H, Kohr G. A boundary Schwarz lemma for mappings from the unit polydisc to irreducible bounded symmetric domains. Math Nachr, 2020, 293(7): 1345-1351
[12] Hamada H, Kohr G.A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains. Math Nachr, 2021, 294(11): 2151-2159
[13] He L, Tu Z. The Schwarz lemma at the boundary of the non-convex complex ellipsoids. Acta Mathematica Scientia, 2019, 39B(4): 915-926
[14] Huang X. A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains. Canad J Math, 1995, 47(2): 405-420
[15] Huang X. A preservation principle of extremal mappings near a strongly pseudoconvex point and its applications. Ill J Math, 1994, 38(2): 283-302
[16] Huang X. On a semi-rigidity property for holomorphic maps. Asian J Math, 2003, 7(4): 463-492
[17] Kim K, Lee H.Schwarz's Lemma from a Differential Geometric Viewpoint. Singapore: World Scientific, 2011
[18] Krantz S.Function Theory of Several Complex Variables. Providence, RI: Amer Math Soc, 2001
[19] Liu T, Tang X. Schwarz lemma at the boundary of strongly pseudoconvex domain in $\mathbb{C}^n$. Math Ann, 2016, 366: 655-666
[20] Liu T, Tang X. A boundary Schwarz lemma on the classical domain of type $\mathcal{I}$. Sci China Math, 2017, 60(7): 1239-1258
[21] Liu T, Tang X. Schwarz lemma and rigidity theorem for holomorphic mappings on the unit polydisk in ${\mathbb{C}}^n$. J Math Anal Appl, 2020, 489(2): 124148
[22] Liu T, Tang X, Zhang W. Schwarz lemma at the boundary on the classical domain of type $\mathcal{III}$. Chin Ann Math Ser B, 2020, 41(3): 335-360
[23] Liu T, Ren G. The growth theorem of convex mappings on bounded convex circular domains. Sci China Math, 1998, 41(2): 123-130
[24] Liu T, Wang J, Tang X.Schwarz lemma at the boundary of the unit ball in ${\mathbb{C}}^n$ and its applications. J Geom Anal, 2015, 25: 1890-1914
[25] Tang X, Liu T, Zhang W. Schwarz lemma at the boundary on the classical domain of type $\mathcal{II}$. J Geom Anal, 2018, 28(2): 1610-1634
[26] Tang X, Liu T, Zhang W. Schwarz lemma at the boundary and rigidity property for holomorphic mappings on the unit ball of ${\mathbb{C}}^n$. Proc Amer Math Soc, 2017, 145: 1709-1716
[27] Wang J, Liu T, Tang X. Schwarz lemma at the boundary on the classical domain of type $\mathcal{IV }$. Pacific J Math, 2019, 302: 309-333
[28] Wang X, Ren G. Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains. Complex Anal Oper Theory, 2017, 11(2): 345-358
[29] Yau S. A general Schwarz lemma for Kähler manifolds. Amer J Math, 1978, 100(1): 197-203
[30] Zimmer A. Two boundary rigidity results for holomorphic maps. Amer J Math, 2022, 144(1): 119-168