Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PROJECTIVE DOMAINS WITH NON-COMPACT AUTOMORPHISM GROUPS I

  • Yi, Chang-Woo (Department of Mathematical Sciences Seoul National University)
  • Published : 2008.09.30
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Abstract

Most of domains people have studied are convex bounded projective (or affine) domains. Edith $Soci{\acute{e}}$-$M{\acute{e}}thou$ [15] characterized ellipsoid in ${\mathbb{R}}^n$ by studying projective automorphism of convex body. In this paper, we showed convex and bounded projective domains can be identified from local data of their boundary points using scaling technique developed by several mathematicians. It can be found that how the scaling technique combined with properties of projective transformations is used to do that for a projective domain given local data around singular boundary point. Furthermore, we identify even unbounded or non-convex projective domains from its local data about a boundary point.

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References

  1. E. Bedford and S. Pinchuk, Domains in $C^{2}$ with noncompact groups of holomorphic automorphisms, Mat. Sb. (N.S.) 135(177) (1988), no. 2, 147-157, 271; translation in Math. USSR-Sb. 63 (1989), no. 1, 141-151
  2. E. Bedford and S. Pinchuk, Domains in $C^{n+1}$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), no. 3, 165-191 https://doi.org/10.1007/BF02921302
  3. J. P. Benzecri, Sur les varietes localement affines et localement projectives, Bull. Soc. Math. France 88 (1960), 229-332
  4. B. Colbois and P. Verovic, Rigidity of Hilbert metrics, Bull. Austral. Math. Soc. 65 (2002), no. 1, 23-34 https://doi.org/10.1017/S0004972700020025
  5. W. M. Goldman, Two examples of affine manifolds, Pacific J. Math. 94 (1981), no. 2, 327-330 https://doi.org/10.2140/pjm.1981.94.327
  6. W. M. Goldman, Geometric structures on manifolds and varieties of representations, Geometry of group representations (Boulder, CO, 1987), 169-198, Contemp. Math., 74, Amer. Math. Soc., Providence, RI, 1988 https://doi.org/10.1090/conm/074/957518
  7. W. M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), no. 3, 791-845 https://doi.org/10.4310/jdg/1214444635
  8. H. Kim, Actions of infinite discrete groups of projective transformation, Proc. Second GARC Symposium on Pure and Applied Math. Part III, 1-9
  9. K. T. Kim, Complete localization of domains with noncompact automorphism groups, Trans. Amer. Math. Soc. 319 (1990), no. 1, 139-153 https://doi.org/10.2307/2001339
  10. K. T. Kim, Geometry of bounded domains and the scaling techniques in several complex variables, Lecture Notes Series 13, Research Institutes of Mathematics Global Analysis Research Center, Seoul National University
  11. S. Kobayashi, Projectively invariant distances for affine and projective structures, Differential geometry (Warsaw, 1979), 127-152, Banach Center Publ., 12, PWN, Warsaw, 1984
  12. N. H. Kuiper, On convex locally-projective spaces, Convegno Internazionale di Geometria Differenziale, Italia, 1953, pp. 200-213. Edizioni Cremonese, Roma, 1954
  13. K. S. Park, Some results on the geometry and topology of affine flat manifolds, Ph. D. thesis, Seoul National University, Aug, 1997
  14. S. Pinchuk, Holomorphic inequivalences of some classes of domains in $\mathbb{C}^{n}$, Mat. USSR Sbornik 39 (1981), 61-86 https://doi.org/10.1070/SM1981v039n01ABEH001470
  15. E. Socie-Methou, Caracterisation des ellipsoides par leurs groupes d'automorphismes, Ann. Sci. Ecole Norm. Sup. (4) 35 (2002), no. 4, 537-548 https://doi.org/10.1016/S0012-9593(02)01103-5
  16. J. Vey, Sur les automorphismes affines des ouverts convexes saillants, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), 641-665
  17. E. B. Vinberg and V. G. Kac, Quasi-homogeneous cones, Math. Notes 1 (1967), 231-235, (translated from Mathematicheskie Zametki, Vol. 1 (1967), no. 3, 347-354
  18. C. W. Yi, Projective domains with non-compact automorphism groups II, preprint

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