The fixed points of holomorphic maps on a convex domain

Do Duc Thai

Annales Polonici Mathematici (1992)

  • Volume: 56, Issue: 2, page 143-148
  • ISSN: 0066-2216

Abstract

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We give a simple proof of the result that if D is a (not necessarily bounded) hyperbolic convex domain in ℂ n then the set V of fixed points of a holomorphic map f:D → D is a connected complex submanifold of D; if V is not empty, V is a holomorphic retract of D. Moreover, we extend these results to the case of convex domains in a locally convex Hausdorff vector space.

How to cite

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Do Duc Thai. "The fixed points of holomorphic maps on a convex domain." Annales Polonici Mathematici 56.2 (1992): 143-148. <http://eudml.org/doc/262465>.

@article{DoDucThai1992,
abstract = {We give a simple proof of the result that if D is a (not necessarily bounded) hyperbolic convex domain in $ℂ^n$ then the set V of fixed points of a holomorphic map f:D → D is a connected complex submanifold of D; if V is not empty, V is a holomorphic retract of D. Moreover, we extend these results to the case of convex domains in a locally convex Hausdorff vector space.},
author = {Do Duc Thai},
journal = {Annales Polonici Mathematici},
keywords = {fixed points; holomorphic maps; convex domains},
language = {eng},
number = {2},
pages = {143-148},
title = {The fixed points of holomorphic maps on a convex domain},
url = {http://eudml.org/doc/262465},
volume = {56},
year = {1992},
}

TY - JOUR
AU - Do Duc Thai
TI - The fixed points of holomorphic maps on a convex domain
JO - Annales Polonici Mathematici
PY - 1992
VL - 56
IS - 2
SP - 143
EP - 148
AB - We give a simple proof of the result that if D is a (not necessarily bounded) hyperbolic convex domain in $ℂ^n$ then the set V of fixed points of a holomorphic map f:D → D is a connected complex submanifold of D; if V is not empty, V is a holomorphic retract of D. Moreover, we extend these results to the case of convex domains in a locally convex Hausdorff vector space.
LA - eng
KW - fixed points; holomorphic maps; convex domains
UR - http://eudml.org/doc/262465
ER -

References

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  2. [2] T. Barth, Convex domains and Kobayashi hyperbolicity, ibid. 79 (1980), 556-558. Zbl0438.32013
  3. [3] S. Dineen, R. Timoney et J.-P. Vigué, Pseudodistances invariantes sur les domaines d'un espace localement convexe, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985), 515-529. Zbl0603.46052
  4. [4] R. E. Edwards, Functional Analysis, Holt, Rinehart and Winston, New York 1965. Zbl0182.16101
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  6. [6] T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland Math. Stud. 40, Amsterdam 1980. Zbl0447.46040
  7. [7] P. Kiernan, On the relations between taut, tight and hyperbolic manifolds, Bull. Amer. Math. Soc. 76 (1970), 49-51. Zbl0192.44103
  8. [8] J. L. Kelley, General Topology, Van Nostrand, New York 1957. 
  9. [9] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Dekker, New York 1970. Zbl0207.37902
  10. [10] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. 
  11. [11] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), 257-261. Zbl0509.32015
  12. [12] H. L. Royden and P. Wong, Carathéodory and Kobayashi metrics on convex domains, to appear. 
  13. [13] E. Vesentini, Complex geodesics, Compositio Math. 44 (1981), 375-394. Zbl0488.30015
  14. [14] E. Vesentini, Complex geodesics and holomorphic maps, in: Sympos. Math. 26, Inst. Naz. Alta Mat. Fr. Severi, 1982, 211-230. Zbl0506.32008
  15. [15] J.-P. Vigué, Points fixes d’applications holomorphes dans un domaine borné convexe de ℂ n , Trans. Amer. Math. Soc. 289 (1985), 345-353. Zbl0589.32043

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