Bulletin of the American Mathematical Society

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Book Information:

Author: Matti Vuorinen
Title: Conformal geometry and quasiregular mappings
Additional book information: Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, Heidelberg, New York, 1988, xix + 209 pp., $21.20. ISBN 3-540-19342-1.

Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038

Lars V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429. MR 167618, DOI 10.2307/2373173

Lars V. Ahlfors, Quasiconformal mappings, Teichmüller spaces, and Kleinian groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 71–84. MR 562598

Kari Astala, A remark on quasiconformal mappings and BMO-functions, Michigan Math. J. 30 (1983), no. 2, 209–212. MR 718266, DOI 10.1307/mmj/1029002851

Albert Baernstein II and Juan J. Manfredi, Topics in quasiconformal mapping, Topics in modern harmonic analysis, Vol. I, II (Turin/Milan, 1982) Ist. Naz. Alta Mat. Francesco Severi, Rome, 1983, pp. 819–862. MR 748884

Lipman Bers, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215–228. MR 132175, DOI 10.1002/cpa.3160140304

Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73–98. MR 477161, DOI 10.1007/BF02545743

Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300. MR 348097, DOI 10.1112/blms/4.3.257

Lipman Bers, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1083–1100. MR 463433, DOI 10.1090/S0002-9904-1977-14390-5

Lipman Bers, Finite-dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 131–172. MR 621883, DOI 10.1090/S0273-0979-1981-14933-8

A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, DOI 10.1007/BF02392360

B. V. Boyarskiĭ, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43(85) (1957), 451–503 (Russian). MR 0106324

B. Bojarski and T. Iwaniec, Another approach to Liouville theorem, Math. Nachr. 107 (1982), 253–262. MR 695751, DOI 10.1002/mana.19821070120

B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $\textbf {R}^{n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. MR 731786, DOI 10.5186/aasfm.1983.0806

David Drasin, The inverse problem of the Nevanlinna theory, Acta Math. 138 (1976), no. 1-2, 83–151. MR 585644, DOI 10.1007/BF02392314

Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 276999

I. S. Gál, Conformally invariant metrics and uniform structures. I. II, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 218–231, 232–244. MR 0118828

F. W. Gehring, Rings and quasiconformal mappings in space, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 98–105. MR 125964, DOI 10.1073/pnas.47.1.98

F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268

Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034

Mariano Giaquinta and Enrico Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46. MR 666107, DOI 10.1007/BF02392725

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0

S. Granlund, P. Lindqvist, and O. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc. 277 (1983), no. 1, 43–73. MR 690040, DOI 10.1090/S0002-9947-1983-0690040-4

S. Granlund, P. Lindqvist, and O. Martio, $F$-harmonic measure in space, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 233–247. MR 686642, DOI 10.5186/aasfm.1982.0717

S. Granlund, P. Lindqvist, and O. Martio, Note on the PWB-method in the nonlinear case, Pacific J. Math. 125 (1986), no. 2, 381–395. MR 863533

S. Granlund, P. Lindqvist, and O. Martio, Phragmén-Lindelöf’s and Lindelöf’s theorems, Ark. Mat. 23 (1985), no. 1, 103–128. MR 800175, DOI 10.1007/BF02384420

Irwin Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), no. 3-4, 231–270. MR 611385, DOI 10.1007/BF02392465

Jacqueline Lelong-Ferrand, Transformations conformes et quasi-conformes des variétés riemanniennes compactes (démonstration de la conjecture de A. Lichnerowicz), Acad. Roy. Belg. Cl. Sci. Mém. Collect. 8$^\textrm {o}$ (2) 39 (1971), no. 5, 44 (French). MR 322739

Jacqueline Lelong-Ferrand, Invariants conformes globaux sur les variétés riemanniennes, J. Differential Geometry 8 (1973), 487–510 (French). MR 346702

Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0

Olli Lehto, A historical survey of quasiconformal mappings, On the work of Leonhard Euler (Berlin, 1983) Birkhäuser, Basel, 1984, pp. 205–217. MR 753330

P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), no. 3-4, 153–171. MR 806413, DOI 10.1007/BF02392541

O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463

A. Marden and S. Rickman, Holomorphic mappings of bounded distortion, Proc. Amer. Math. Soc. 46 (1974), 226–228. MR 348146, DOI 10.1090/S0002-9939-1974-0348146-5

O. Martio and S. Rickman, Measure properties of the branch set and its image of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 541 (1973), 16. MR 352453

O. Martio, S. Rickman, and J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 465 (1970), 13. MR 0267093

O. Martio, S. Rickman, and J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 465 (1970), 13. MR 0267093

O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 488 (1971), 31. MR 299782

G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383

Ju. G. Rešetnjak, Liouville’s conformal mapping theorem under minimal regularity hypotheses, Sibirsk. Mat. Ž. 8 (1967), 835–840 (Russian). MR 0218544

Ju. G. Rešetnjak, Spatial mappings with bounded distortion, Sibirsk. Mat. Ž. 8 (1967), 629–658 (Russian). MR 0215990

Ju. G. Rešetnjak, Mappings with bounded distortion as extremals of integrals of Dirichlet type, Sibirsk. Mat. Ž. 9 (1968), 652–666 (Russian). MR 0230900

Yu. G. Reshetnyak, Prostranstvennye otobrazheniya s ogranichennym iskazheniem, “Nauka” Sibirsk. Otdel., Novosibirsk, 1982 (Russian). MR 665590

Seppo Rickman, A defect relation for quasimeromorphic mappings, Ann. of Math. (2) 114 (1981), no. 1, 165–191. MR 625350, DOI 10.2307/1971382

Seppo Rickman, The analogue of Picard’s theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), no. 3-4, 195–242. MR 781587, DOI 10.1007/BF02392472

Hans Martin Reimann and Thomas Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Mathematics, Vol. 487, Springer-Verlag, Berlin-New York, 1975 (German). MR 0511997

Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, DOI 10.2307/1971308

Jussi Väisälä, On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I No. 298 (1961), 36. MR 0140685

[V2] J. Väisälä, Quasiregular mappings in n-space, XVI, Scandinavian Congress of Mathematicians, Göteburg, Sweden, 1972.

Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009

V. A. Zorič, M. A. Lavrent′ev’s theorem on quasiconformal space maps, Mat. Sb. (N.S.) 74 (116) (1967), 417–433 (Russian). MR 0223569

[A1] L. Ahlfors, Lectures on quasiconformal mappings, Wadsworth & Brooks/Cole Mathematical Series, 1987. MR 2241787




[A2] L. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413-429. MR 167618




[A3] L. Ahlfors, Quasiconformal mappings, Teichmüller spaces, and Kleinian groups, Proceedings of the International Congress of Mathematicians, Helsinki, 1978, pp. 71-84. MR 562598




[As] K. Astala, A remark on quasi-conformal mappings and BMO-functions, Michigan Math. J. 30 (1983), 209-212. MR 718266




[BM] A. Baernstein and J. Manfredi, Topics in quasiconformal mappings, Topics in modern analysis, (Leo de Michelle, ed.) Istituto Nazionale di Alta Matematica, Roma, 1983, pp. 819-862. MR 748884




[B1] L. Bers, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215-228. MR 132175




[B2] L. Bers, An extremal problem for quasiconformal mappings and a problem of Thurston, Acta Math. 141 (1978), 73-98. MR 477161




[B3] L. Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257-300. MR 348097




[B4] L. Bers, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc. 83 (1977), 1083-1100. MR 463433




[B5] L. Bers, Finite dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 131-172. MR 621883




[BA] A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125-142. MR 86869




[Bo] B. Bojarski, Generalized solutions of a system of first order differential equations of elliptic type with discontinuous coefficients, Mat. Sb. 43 (1957), 451-503 (Russian). MR 106324




[BI1] B. Bojarski and T. Iwaniec, Another approach to Liouville theorem, Math. Nachr. 107 (1982), 253-262. MR 695751




[BI2] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in R, Ann. Acad. Sci. Fenn. Ser A I Math. 8 (1983), 257-324. MR 731786




[D] D. Drasin, The inverse problem of the Nevanlinna theory, Acta Math. 138 (1977), 83-151. MR 585644




[EE] C. Earle and J. Eells, A fibre bundle description of Teichmüller theory, J. Differential Geom. 3 (1969), 19-43. MR 276999




[Ga] I. Gál, Conformally invariant metrics and uniform structures, Indag. Math. 22 (1960), 218-244. MR 118828




[G1] F. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 101 (1961), 499-519. MR 125964




[G2] F. Gehring, The L, Acta Math. 130 (1973), 265-277. MR 402038




[G] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Studies 105, Princeton Univ. Press, Princeton, N. J., 1983. MR 717034




[GG] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46. MR 666107




[GT] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin and New York, 1983. MR 737190




[GLM1] S. Granlund, P. Lindqvist and O. Martio, Conformally invariant variational integrals, Trans. Amer. Math. Soc. 277 (1983), 43-47. MR 690040




[GLM2] S. Granlund, P. Lindqvist and O. Martio, F-harmonic measure in space, Ann. Acad. Sci. Fenn. A I Math. 7 (1982), 233-247. MR 686642




[GLM3] S. Granlund, P. Lindqvist and O. Martio, Note on the PWB-method in the non-linear case, Pacific J. Math. 125 (1986), 381-395. MR 863533




[GLM4] S. Granlund, P. Lindqvist and O. Martio, Phragmén-Lindeöf's and Lindelöf's theorems, Ark. Math. 23 (1985), 103-128. MR 800175




[K] I. Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), 231-270. MR 611385




[LF1] J. Lelong-Ferrand, Transformations conformes et quasi-conformes des variétés riemanniennes compactes (Démonstration de la conjecture de A. Lichnerowicz), Acad. Roy. Belg. Cl. Sci. Mém. Coll. 39 (1971), 1-44. MR 322739




[LF2] J. Lelong-Ferrand, Invariants conformes globaux sur les variétés riemanniennes, J. Differential Geom. 8 (1973), 487-510. MR 346702




[L1] O. Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Math., vol. 109, Springer-Verlag, Berlin and New York, 1987. MR 867407




[L2] O. Lehto, A historical survey of quasiconformal mappings, Lectures from the Euler Colloquium held at the Technische Universität, Berlin, May 12-14, 1983; (E. Knoblich, I. S. Louhivaara and J. Winkler, eds.) Birkhäuser, Basel-Boston, 1984, pp. 205-217. MR 753330




[LM] P. Lindqvist and O. Martio, Two theorems of N. Wiener for solutions of quasilinear elliptic equations, Acta Math. 155 (1985), 153-171. MR 806413




[LV] O. Lehto and K. Virtanen, Quasi-conformal mappings in the plane, 2nd ed., Springer-Verlag, Berlin and New York, 1973. MR 344463




[MaR] A. Marden and S. Rickman, Holomorphic mappings of a bounded distortion, Proc. Amer. Math. Soc. 46 (1974), 226-228. MR 348146




[MR] A. Marden and S. Rickman, Measure properties of the branch set and its image of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 541 (1973), 1-16. MR 352453




[MRV1] O. Martio, S. Rickman and J. Väisälä, Definitions for quasi-regular mappings, Ann. Acad. Sci. Fenn. Ser A I 465 (1970), 1-13. MR 267093




[MRV2] O. Martio, S. Rickman and J. Väisälä, Distorsion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser A I 465 (1970), 1-13. MR 267093




[MRV3] O. Martio, S. Rickman and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn Ser A I 488 (1971), 1-31. MR 299782




[M] G. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Publ. Math. Inst. Hautes Études Sci. 34 (1968), 53-104. MR 236383




[R1] J. Rešetnjak, The Liouville theorem with minimal regularity conditions, Sibirsk. Mat. Z. 8 (1967), 835-840. MR 218544




[R2] J. Rešetnjak, Space mappings with bounded distortion, Sibirsk. Mat. Z. 8 (1967), 629-658. (Russian) MR 215990




[R3] J. Rešetnjak, Mappings with bounded distortion as extremals of Dirichlet type integrals, Sibirsk. Mat. Z. 9 (1968), 652-666. (Russian) MR 230900




[R4] J. Rešetnjak, Spatial mappings with bounded distortion, Izdat. "Nauka", Sibirsk. Otdeleine, Novosibirsk, 1982. (Russian) MR 665590




[Ri1] S. Rickman, A defect relation for quasimeromorphic mappings, Ann. of Math. (2) 114 (1981), 165-191. MR 625350




[Ri2] S. Rickman, The analogue of Picard's theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), 195-242. MR 781587




[RR] H. Reiman and T. Rychener, Funktionen mit beschränkter mittlerer Oszillation, Lecture Notes in Math., vol. 487, Springer-Verlag, Berlin and New York, 1975. MR 511997




[S] D. Sullivan, Quasiconformal homeomorphisms and dynamics. I, Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), 401-418. MR 819553




[V1] J. Väisälä, On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. 298 (1961), 1-36. MR 140685




[V2] J. Väisälä, Quasiregular mappings in n-space, XVI, Scandinavian Congress of Mathematicians, Göteburg, Sweden, 1972.




[V3] J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin and New York, 1971. MR 454009




[Z] V. A. Zǒric, The theorem of M. A. Laurentiev on quasiconformal mappings in space, Mat. Sb. 74 (1967), 417-433. (Russian) MR 223569