Growth of proper holomorphic maps and tropical power series

Evgeny Abakumov; Evgueni Doubtsov

doi:10.1016/j.crma.2016.03.001

Complex analysis

Growth of proper holomorphic maps and tropical power series

Presented by: Jean-Pierre Demailly

Evgeny Abakumov ¹; Evgueni Doubtsov ^2,³

¹ Université Paris-Est, LAMA (UMR 8050), 77454 Marne-la-Vallée, France
² St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
³ Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr. 28, St. Petersburg 198504, Russia

Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 465-469.

Abstract
Résumé

Motivated by a question of J. Globevnik, we show that a proper holomorphic immersion of the unit disk $D$ into $C^{2}$ or a proper holomorphic embedding $f : D \to C^{3}$ may have arbitrary growth. Also, using tropical power series, we characterize those radial weights w on the complex plane for which there exist $n \in N$ and a proper holomorphic map $f : C \to C^{n}$ such that $| f (z) |$ is equivalent to $w (z)$ .

Motivés par une question de J. Globevnik, nous montrons qu'une immersion holomorphe propre du disque unité $D$ dans $C^{2}$ ou un plongement holomorphe propre $f : D \to C^{3}$ peut avoir une croissance arbitraire. En outre, en utilisant les séries entières tropicales, nous caractérisons les poids radiaux w sur le plan complexe pour lesquels il existe $n \in N$ et une application holomorphe propre $f : C \to C^{n}$ tels que $| f (z) |$ soit équivalente à $w (z)$ .

Received: 2016-01-12
Accepted: 2016-03-03
Published online: 2016-03-29

DOI: 10.1016/j.crma.2016.03.001

Author's affiliations:

Evgeny Abakumov ¹; Evgueni Doubtsov ^{2, 3}

¹ Université Paris-Est, LAMA (UMR 8050), 77454 Marne-la-Vallée, France
² St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
³ Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr. 28, St. Petersburg 198504, Russia

@article{CRMATH_2016__354_5_465_0,
     author = {Evgeny Abakumov and Evgueni Doubtsov},
     title = {Growth of proper holomorphic maps and tropical power series},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {465--469},
     publisher = {Elsevier},
     volume = {354},
     number = {5},
     year = {2016},
     doi = {10.1016/j.crma.2016.03.001},
     language = {en},
}

TY  - JOUR
AU  - Evgeny Abakumov
AU  - Evgueni Doubtsov
TI  - Growth of proper holomorphic maps and tropical power series
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 465
EP  - 469
VL  - 354
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2016.03.001
LA  - en
ID  - CRMATH_2016__354_5_465_0
ER  -

%0 Journal Article
%A Evgeny Abakumov
%A Evgueni Doubtsov
%T Growth of proper holomorphic maps and tropical power series
%J Comptes Rendus. Mathématique
%D 2016
%P 465-469
%V 354
%N 5
%I Elsevier
%R 10.1016/j.crma.2016.03.001
%G en
%F CRMATH_2016__354_5_465_0

Evgeny Abakumov; Evgueni Doubtsov. Growth of proper holomorphic maps and tropical power series. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 465-469. doi : 10.1016/j.crma.2016.03.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.001/

References
Cited by

[1] E. Abakumov; E. Doubtsov Reverse estimates in growth spaces, Math. Z., Volume 271 (2012) no. 1–2, pp. 399-413

[2] E. Abakumov; E. Doubtsov Moduli of holomorphic functions and logarithmically convex radial weights, Bull. Lond. Math. Soc., Volume 47 (2015) no. 3, pp. 519-532

[3] H. Alexander Explicit imbedding of the (punctured) disc into $C^{2}$ , Comment. Math. Helv., Volume 52 (1977) no. 4, pp. 539-544

[4] K.D. Bierstedt; J. Bonet; J. Taskinen Associated weights and spaces of holomorphic functions, Stud. Math., Volume 127 (1998) no. 2, pp. 137-168

[5] J. Bonet; P. Domański; M. Lindström Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Can. Math. Bull., Volume 42 (1999) no. 2, pp. 139-148

[6] E. Doubtsov Harmonic approximation by finite sums of moduli, J. Math. Anal. Appl., Volume 430 (2015) no. 2, pp. 685-694

[7] Y. Eliashberg; M. Gromov Embeddings of Stein manifolds of dimension n into the affine space of dimension $3 n / 2 + 1$ , Ann. Math. (2), Volume 136 (1992) no. 1, pp. 123-135

[8] P. Erdös; T. Kövári On the maximum modulus of entire functions, Acta Math. Acad. Sci. Hung., Volume 7 (1956), pp. 305-317

[9] F. Forstnerič; E.F. Wold Embeddings of infinitely connected planar domains into $C^{2}$ , Anal. PDE, Volume 6 (2013) no. 2, pp. 499-514

[10] J. Globevnik On growth of holomorphic embeddings into $C^{2}$ , Proc. R. Soc. Edinb., Sect. A, Volume 132 (2002) no. 4, pp. 879-889

[11] J. Globevnik; B. Stensønes Holomorphic embeddings of planar domains into $C^{2}$ , Math. Ann., Volume 303 (1995) no. 4, pp. 579-597

[12] I. Itenberg; G. Mikhalkin; E. Shustin Tropical Algebraic Geometry, Oberwolfach Seminars, vol. 35, Birkhäuser Verlag, Basel, Switzerland, 2009

[13] K. Kasahara; T. Nishino Math. Rev., 38 (1969) (As announced in)

[14] C.O. Kiselman Croissance des fonctions plurisousharmoniques en dimension infinie, Ann. Inst. Fourier (Grenoble), Volume 34 (1984) no. 1, pp. 155-183

[15] C.O. Kiselman Questions inspired by Mikael Passare's mathematics, Afr. Math., Volume 25 (2014) no. 2, pp. 271-288

[16] P. Koosis Sur l'approximation pondérée par des polynômes et par des sommes d'exponentielles imaginaires, Ann. Sci. Éc. Norm. Super. (3), Volume 81 (1964), pp. 387-408

[17] H.B. Laufer Imbedding annuli in $C^{2}$ , J. Anal. Math., Volume 26 (1973), pp. 187-215

[18] D.S. Lubinsky Strong Asymptotics for Extremal Errors and Polynomials Associated with Erdős-Type Weights, Pitman Research Notes in Mathematics Series, vol. 202, Longman Scientific & Technical, Harlow, 1989

[19] S.N. Mergelyan (American Mathematical Society Translations, Ser. 2), Volume vol. 10, American Mathematical Society, Providence, RI, USA (1958), pp. 59-106

[20] U. Schmid On the approximation of positive functions by power series. II, J. Approx. Theory, Volume 92 (1998) no. 3, pp. 486-501

[21] J.-L. Stehlé Plongements du disque dans $C^{2}$ , Séminaire Pierre-Lelong (Analyse), Année 1970–1971, Lecture Notes in Math., vol. 275, Springer, Berlin, 1972, pp. 119-130

Cited by Sources:

Comments - Politique