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Acta Mathematica Sinica, English Series

, Volume 34, Issue 8, pp 1278–1288 | Cite as

Minimum Principle for Plurisubharmonic Functions and Related Topics

  • Fu Sheng Deng
  • Hui Ping Zhang
  • Xiang Yu Zhou
Article
  • 28 Downloads

Abstract

This is a survey about some recent developments of the minimum principle for plurisubharmonic functions and related topics.

Keywords

Minimum principle plurisubharmonic functions Stein manifolds geometric invariant theory group actions holomorphic vector bundles 

MR(2010) Subject Classification

32M05 32U05 

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References

  1. [1]
    Berman, R. J., Berndtsson, B.: The volume of Kähler–Einstein Fano varieties and convex bodies. J. Reine Angew. Math., 723, 127–152 (2017)MathSciNetzbMATHGoogle Scholar
  2. [2]
    Berndtsson, B.: Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions. Math. Ann., 312, 785–792 (1998) geom 81.3, pp. 457–482 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. of Math. (2), 169(2), 531–560 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bröcker, T., Dieck, T. T.: Representations of Compact Lie Groups, GTM98, Springer-Verlag, 1985CrossRefzbMATHGoogle Scholar
  5. [5]
    Burns, D., Halverscheid, S., Hind, R.: The Geometry of Grauert Tubes and Complexification of Symmetric Spaces. Duke Math. J., 118, 465–491 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Chafi, B.: Principe du Minimum pour les fonctions plurisousharmoniques. Thèse de 3e cycle, Université de Lille 1, (juin 1983)Google Scholar
  7. [7]
    Demailly, J. P.: Complex Analytic and Differential Geometry, e-book, available at: https://doi.org/www-fourier.ujfgrenoble.fr/demailly/documents.html
  8. [8]
    Deng, F., Rong, F.: On biholomorphisms between bounded quasi-Reinhardt domains. Annali di Matematica Pura ed Applicata, 195, 835–843 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Deng, F., Zhang, H., Zhou, X.: Positivity of direct images of positively curved volume forms. Math. Z., 278, 347–362 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Deng, F., Zhang, H., Zhou, X.: Positivity of character subbundles and minimumprinciple for noncompact group actions. Math. Z., 286, 431–442 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Fels, G., Huckleberry, A., Wolf, J. A.: Cycle Spaces of Flag Domains, Progressin Math. 245, Birkhäuser, Boston, 2006zbMATHGoogle Scholar
  12. [12]
    Guan, Q., Zhou, X.: A solution of an L 2-extension problem with an optimal estimate and applications. Ann. of Math. (2), 181(3), 1139–1208 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann., 281, 631–662 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Heinzner, P.: The minimum principle from a Hamiltonian point of view. Doc. Math. J. DMV, 3, 1–14 (1998)MathSciNetzbMATHGoogle Scholar
  15. [15]
    Hitchin, N. J., Karlhede, A., Lindström, U., et al.: HyperKähler metrics and supersymmetry. Commun. Math. Phys., 108, 535–589 (1987)CrossRefzbMATHGoogle Scholar
  16. [16]
    Kiselman, C.: The partial Legendre transformation for plurisubharmonic functions. Invent. Math., 49(2), 137–148 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Kiselman, C.: Densité des fonctions plurisousharmoniques. Bull. Soc. Math. France, 107, 295–304 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Kiselman, C.: La teoremo de Siu por abstraktaj nombroj de Lelong. Aktoj de Internacia Scienca Akademio Comenius, 1, 56–65 (1992)MathSciNetGoogle Scholar
  19. [19]
    Kiselman, C.: Attenuating the singularities of plurisubharmonic functions. Ann. Polon. Math., 60, 173–197 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Krotz, B., Stanton, J.: Holomorphic extensions of representations: (I) automorphic functions. Ann. of Math., 150, 641–724 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Krotz, B., Stanton, J.: Holomorphic extensions of representations: (II) Geometry and harmonic analysis. Geom. Funct. Anal., 15, 190–245 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Loeb, J. J.: Action d’une forme réelle d’un groupe de Lie complexe sur les fonctions plurisousharmoniques. Ann. Inst. Fourier, 35, 59–97 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Onishchik, A. L., Vinberg, E. B.: Lie Groups and Lie Algebras II: Discrete Subgroups of Lie Groups and Cohomologies of Lie Groups and Lie Algebras, Springer-Verlag, Berlin Heidelberg, 2000zbMATHGoogle Scholar
  24. [24]
    Prekopa, A.: On logarithmic concave measures and functions. Acad Sci. Math. (Szeged), 34, 335–343 (1973)MathSciNetzbMATHGoogle Scholar
  25. [25]
    Snow, D. M.: Reductive group action on Stein spaces. Math. Ann., 259, 79–97 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Zhou, X.: A proof of the extended future tube conjecture. Izvestiya Ran, Series Math. T., 62, 211–224 (1998)MathSciNetGoogle Scholar
  27. [27]
    Zhou, X. Y.: Some results related to group actions in several complex variables. Proc. of ICM 2002, 2, 743–753 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Fu Sheng Deng
    • 1
  • Hui Ping Zhang
    • 2
  • Xiang Yu Zhou
    • 3
  1. 1.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingP. R. China
  2. 2.Department of Mathematics, School of InformationRenmin (People’s) University of ChinaBeijingP. R. China
  3. 3.Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingP. R. China

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