The Julia–Wolff–Carathéodory theorem in convex finite type domains

Leandro Arosio; Matteo Fiacchi

doi:10.1515/crelle-2025-0041

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The Julia–Wolff–Carathéodory theorem in convex finite type domains

Leandro Arosio und Matteo Fiacchi

Veröffentlicht/Copyright: 22. Juli 2025

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2025 Heft 827

Abstract

Rudin’s version of the classical Julia–Wolff–Carathéodory theorem is a cornerstone of holomorphic function theory in the unit ball of C d . In this paper, we obtain a complete generalization of Rudin’s theorem for a holomorphic map f : D → D ′ between convex domains of finite type. In particular, given a point ξ ∈ ∂ D with finite dilation, we show that the 𝐾-limit of 𝑓 at 𝜉 exists and is a point η ∈ ∂ D ′ , and we obtain asymptotic estimates for all entries of the Jacobian matrix of the differential d ⁢ f z in terms of the multitypes at the points 𝜉 and at 𝜂. We introduce a generalization of Bracci–Patrizio–Trapani’s pluricomplex Poisson kernel which, together with the dilation at 𝜉, gives a formula for the restricted 𝐾-limit of the normal component of the normal derivative ⟨ d ⁢ f z ⁢ ( n ξ ) , n η ⟩ . Our principal tools are methods from Gromov hyperbolicity theory, a scaling in the normal direction, and the strong asymptoticity of complex geodesics. To obtain our main result, we prove a conjecture by Abate on the Kobayashi type of a vector 𝑣, proving that it is equal to the reciprocal of the line type of 𝑣, and we give new extrinsic characterizations of both 𝐾-convergence and restricted convergence to a point ξ ∈ ∂ D in terms of the multitype at 𝜉.

Funding source: Ministero dell’Università e della Ricerca

Award Identifier / Grant number: 2022AP8HZ9

Award Identifier / Grant number: E83C23000330006

Funding source: European Research Council

Award Identifier / Grant number: 101053085

Funding source: Javna Agencija za Raziskovalno Dejavnost RS

Award Identifier / Grant number: P1-0291

Funding statement: Leandro Arosio was partially supported by INdAM, by PRIN Real and Complex Manifolds: Geometry and Holomorphic Dynamics n. 2022AP8HZ9, and by the MUR Excellence Department Project MatMod@TOV CUP: E83C23000330006. Matteo Fiacchi was partially supported by the European Union (ERC Advanced grant HPDR, 101053085 to Franc Forstnerič), the research program P1-0291 from ARIS, Republic of Slovenia, and by the MUR Excellence Department Project MatMod@TOV CUP: E83C23000330006.

Acknowledgements

We want to thank the anonymous referee for the careful reading of the paper and for many useful comments.

References

[1] M. Abate, Common fixed points of commuting holomorphic maps, Math. Ann. 283 (1989), no. 4, 645–655. 10.1007/BF01442858Suche in Google Scholar

[2] M. Abate, Iteration theory of holomorphic maps on taut manifolds, Res. Lecture Notes Math. Complex Anal. Geom., Mediterranean Press, Rende 1989. Suche in Google Scholar

[3] M. Abate, The Lindelöf principle and the angular derivative in strongly convex domains, J. Anal. Math. 54 (1990), 189–228. 10.1007/BF02796148Suche in Google Scholar

[4] M. Abate, Angular derivatives in strongly pseudoconvex domains, Several complex variables and complex geometry, Part 2, Proc. Sympos. Pure Math. 52, American Mathematical Society, Providence (1991), 23–40. 10.1090/pspum/052.2/1128532Suche in Google Scholar

[5] M. Abate, The Julia–Wolff–Carathéodory theorem in polydisks, J. Anal. Math. 74 (1998), 275–306. 10.1007/BF02819453Suche in Google Scholar

[6] M. Abate, Angular derivatives in several complex variables, Real methods in complex and CR geometry, Lecture Notes in Math. 1848, Springer, Berlin (2004), 1–47. 10.1007/978-3-540-44487-9_1Suche in Google Scholar

[7] M. Abate and J. Raissy, A Julia–Wolff–Carathéodory theorem for infinitesimal generators in the unit ball, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5415–5431. 10.1090/tran/6535Suche in Google Scholar

[8] M. Abate and R. Tauraso, The Lindelöf principle and angular derivatives in convex domains of finite type, J. Aust. Math. Soc. 73 (2002), no. 2, 221–250. 10.1017/S1446788700008818Suche in Google Scholar

[9] A. Altavilla, L. Arosio and L. Guerini, Canonical models on strongly convex domains via the squeezing function, J. Geom. Anal. 31 (2021), no. 5, 4661–4702. 10.1007/s12220-020-00448-5Suche in Google Scholar

[10] L. Arosio, F. Bracci and M. Fiacchi, The pluricomplex Poisson kernel for convex domains of finite type, in preparation. Suche in Google Scholar

[11] L. Arosio, G. M. Dall’Ara and M. Fiacchi, Worm domains are not Gromov hyperbolic, J. Geom. Anal. 33 (2023), no. 8, Paper No. 257. 10.1007/s12220-023-01320-ySuche in Google Scholar PubMed PubMed Central

[12] L. Arosio, M. Fiacchi, S. Gontard and L. Guerini, The horofunction boundary of a Gromov hyperbolic space, Math. Ann. 388 (2024), no. 2, 1163–1204. 10.1007/s00208-022-02551-0Suche in Google Scholar

[13] L. Arosio, M. Fiacchi, L. Guerini and A. Karlsson, Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces, Adv. Math. 439 (2024), Article ID 109484. 10.1016/j.aim.2023.109484Suche in Google Scholar

[14] Z. M. Balogh and M. Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000), no. 3, 504–533. 10.1007/s000140050138Suche in Google Scholar

[15] T. J. Barth, Convex domains and Kobayashi hyperbolicity, Proc. Amer. Math. Soc. 79 (1980), no. 4, 556–558. 10.1090/S0002-9939-1980-0572300-3Suche in Google Scholar

[16] G. Bharali and A. Zimmer, Goldilocks domains, a weak notion of visibility, and applications, Adv. Math. 310 (2017), 377–425. 10.1016/j.aim.2017.02.005Suche in Google Scholar

[17] H. P. Boas and E. J. Straube, On equality of line type and variety type of real hypersurfaces in C n , J. Geom. Anal. 2 (1992), no. 2, 95–98. 10.1007/BF02921382Suche in Google Scholar

[18] F. Bracci, M. D. Contreras and S. Díaz-Madrigal, Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 1, 23–53. 10.4171/jems/188Suche in Google Scholar

[19] F. Bracci, H. Gaussier and A. Zimmer, Homeomorphic extension of quasi-isometries for convex domains in C d and iteration theory, Math. Ann. 379 (2021), no. 1–2, 691–718. 10.1007/s00208-020-01954-1Suche in Google Scholar

[20] F. Bracci, N. Nikolov and P. J. Thomas, Visibility of Kobayashi geodesics in convex domains and related properties, Math. Z. 301 (2022), no. 2, 2011–2035. 10.1007/s00209-022-02978-wSuche in Google Scholar

[21] F. Bracci and G. Patrizio, Monge-Ampère foliations with singularities at the boundary of strongly convex domains, Math. Ann. 332 (2005), no. 3, 499–522. 10.1007/s00208-005-0633-7Suche in Google Scholar

[22] F. Bracci, G. Patrizio and S. Trapani, The pluricomplex Poisson kernel for strongly convex domains, Trans. Amer. Math. Soc. 361 (2009), no. 2, 979–1005. 10.1090/S0002-9947-08-04549-2Suche in Google Scholar

[23] F. Bracci and A. Saracco, Hyperbolicity in unbounded convex domains, Forum Math. 21 (2009), no. 5, 815–825. 10.1515/FORUM.2009.039Suche in Google Scholar

[24] F. Bracci, A. Saracco and S. Trapani, The pluricomplex Poisson kernel for strongly pseudoconvex domains, Adv. Math. 380 (2021), Article ID 107577. 10.1016/j.aim.2021.107577Suche in Google Scholar

[25] F. Bracci and D. Shoikhet, Boundary behavior of infinitesimal generators in the unit ball, Trans. Amer. Math. Soc. 366 (2014), no. 2, 1119–1140. 10.1090/S0002-9947-2013-05996-XSuche in Google Scholar

[26] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer, Berlin 1999. 10.1007/978-3-662-12494-9Suche in Google Scholar

[27] C. Carathéodory, Über die Winkelderivierten von beschränkten analytischen Funktionen, Sitzungsber, Preuss. Akad. Wiss. Berlin 1929 (1929), 39–54. Suche in Google Scholar

[28] D. Catlin, Boundary invariants of pseudoconvex domains, Ann. of Math. (2) 120 (1984), no. 3, 529–586. 10.2307/1971087Suche in Google Scholar

[29] D. Catlin, Subelliptic estimates for the ∂ ̄ -Neumann problem on pseudoconvex domains, Ann. of Math. (2) 126 (1987), no. 1, 131–191. 10.2307/1971347Suche in Google Scholar

[30] E. M. Čirka, The Lindelöf and Fatou theorems in C n , Math. USSR Sb. 21 (1973), 619–641. 10.1070/SM1973v021n04ABEH002039Suche in Google Scholar

[31] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Math. 1441, Springer, Berlin 1990. 10.1007/BFb0084913Suche in Google Scholar

[32] J. P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. 10.2307/2007015Suche in Google Scholar

[33] J. P. D’Angelo, Several complex variables and the geometry of real hypersurfaces, Stud. Adv. Math., CRC Press, Boca Raton 2019. 10.1201/9780203739785Suche in Google Scholar

[34] K. Diederich and J. E. Fornaess, Pseudoconvex domains with real-analytic boundary, Ann. of Math. (2) 107 (1978), no. 2, 371–384. 10.2307/1971120Suche in Google Scholar

[35] M. Fiacchi, Gromov hyperbolicity of pseudoconvex finite type domains in C 2 , Math. Ann. 382 (2022), no. 1–2, 37–68. 10.1007/s00208-020-02135-wSuche in Google Scholar

[36] H. Gaussier, Characterization of convex domains with noncompact automorphism group, Michigan Math. J. 44 (1997), no. 2, 375–388. 10.1307/mmj/1029005712Suche in Google Scholar

[37] H. Gaussier and H. Seshadri, On the Gromov hyperbolicity of convex domains in C n , Comput. Methods Funct. Theory 18 (2018), no. 4, 617–641. 10.1007/s40315-018-0243-5Suche in Google Scholar

[38] H. Gaussier and A. Zimmer, The space of convex domains in complex Euclidean space, J. Geom. Anal. 30 (2020), no. 2, 1312–1358. 10.1007/s12220-019-00346-5Suche in Google Scholar

[39] I. Graham, Sharp constants for the Koebe theorem and for estimates of intrinsic metrics on convex domains, Several complex variables and complex geometry, Part 2, Proc. Sympos. Pure Math. 52, American Mathematical Society, Providence (1991), 233–238. 10.1090/pspum/052.2/1128547Suche in Google Scholar

[40] L. A. Harris, Schwarz–Pick systems of pseudometrics for domains in normed linear spaces, Advances in holomorphy, Notas Mat. 65, North-Holland, Amsterdam (1979), 345–406. 10.1016/S0304-0208(08)70766-7Suche in Google Scholar

[41] X. J. Huang, A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains, Canad. J. Math. 47 (1995), no. 2, 405–420. 10.4153/CJM-1995-022-3Suche in Google Scholar

[42] M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis, De Gruyter Exp. Math. 9, Walter de Gruyter, Berlin 2013. 10.1515/9783110253863Suche in Google Scholar

[43] M. Jarnicki, P. Pflug and R. Zeinstra, Geodesics for convex complex ellipsoids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 20 (1993), no. 4, 535–543. Suche in Google Scholar

[44] G. Julia, Extension nouvelle d’un lemme de Schwarz, Acta Math. 42 (1920), no. 1, 349–355. 10.1007/BF02404416Suche in Google Scholar

[45] S. Kobayashi, Hyperbolic complex spaces, Grundlehren Math. Wiss. 318, Springer, Berlin 1998. 10.1007/978-3-662-03582-5Suche in Google Scholar

[46] J. J. Kohn, Subellipticity of the ∂ ̄ -Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math. 142 (1979), no. 1–2, 79–122. 10.1007/BF02395058Suche in Google Scholar

[47] A. Korányi, Harmonic functions on Hermitian hyperbolic space, Trans. Amer. Math. Soc. 135 (1969), 507–516. 10.1090/S0002-9947-1969-0277747-0Suche in Google Scholar

[48] A. Korányi and E. M. Stein, Fatou’s theorem for generalized halfplanes, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 22 (1968), 107–112. Suche in Google Scholar

[49] L. Kosiński, N. Nikolov and A. Y. Ökten, Precise estimates of invariant distances on strongly pseudoconvex domains, Adv. Math. 478 (2025), Article ID 110388. 10.1016/j.aim.2025.110388Suche in Google Scholar

[50] S. G. Krantz, Invariant metrics and the boundary behavior of holomorphic functions on domains in C n , J. Geom. Anal. 1 (1991), no. 2, 71–97. 10.1007/BF02938115Suche in Google Scholar

[51] L. Lee, Asymptotic behavior of the Kobayashi metric on convex domains, Pacific J. Math. 238 (2008), no. 1, 105–118. 10.2140/pjm.2008.238.105Suche in Google Scholar

[52] J. D. McNeal, Convex domains of finite type, J. Funct. Anal. 108 (1992), no. 2, 361–373. 10.1016/0022-1236(92)90029-ISuche in Google Scholar

[53] P. R. Mercer, Complex geodesics and iterates of holomorphic maps on convex domains in C n , Trans. Amer. Math. Soc. 338 (1993), no. 1, 201–211. 10.1090/S0002-9947-1993-1123457-0Suche in Google Scholar

[54] N. Nikolov, A. Y. Ökten and P. J. Thomas, Local and global notions of visibility with respect to Kobayashi distance: A comparison, Ann. Polon. Math. 132 (2024), no. 2, 169–185. 10.4064/ap230522-18-12Suche in Google Scholar

[55] N. Nikolov, P. Pflug and W. Zwonek, Estimates for invariant metrics on ℂ-convex domains, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6245–6256. 10.1090/S0002-9947-2011-05273-6Suche in Google Scholar

[56] J. Raissy, The Julia–Wolff–Carathéodory theorem and its generalizations, Complex analysis and geometry, Springer Proc. Math. Stat. 144, Springer, Tokyo (2015), 281–293. 10.1007/978-4-431-55744-9_21Suche in Google Scholar

[57] R. M. Range, Holomorphic functions and integral representations in several complex variables, Grad. Texts in Math. 108, Springer, New York 1986. 10.1007/978-1-4757-1918-5Suche in Google Scholar

[58] H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II, Lecture Notes in Math. 185, Springer, Berlin (1971), 125–137. 10.1007/BFb0058768Suche in Google Scholar

[59] W. Rudin, Function theory in the unit ball of C n , Grundlehren Math. Wiss. 241, Springer, New York 1980. 10.1007/978-1-4613-8098-6Suche in Google Scholar

[60] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Math. Notes 11, Princeton University, Princeton 1972. Suche in Google Scholar

[61] J. Wolff, Sur une généralisation d’un théorème de Schwarz, C.R. Acad. Sci. Paris 183 (1926), 500–502. Suche in Google Scholar

[62] J. Y. Yu, Multitypes of convex domains, Indiana Univ. Math. J. 41 (1992), no. 3, 837–849. 10.1512/iumj.1992.41.41044Suche in Google Scholar

[63] A. Zimmer, Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type, Math. Ann. 365 (2016), no. 3–4, 1425–1498. 10.1007/s00208-015-1278-9Suche in Google Scholar

[64] A. Zimmer, Subelliptic estimates from Gromov hyperbolicity, Adv. Math. 402 (2022), Article ID 108334. 10.1016/j.aim.2022.108334Suche in Google Scholar

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Revised: 2025-05-22

Published Online: 2025-07-22

Published in Print: 2025-10-01

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