Advertisement

Intersection homology theory via rectifiable currents

  • Qinglan Xia
Article

Abstract.

Here is given a rectifiable currents’ version of intersection homology theory on stratified subanalytic pseudomanifolds. This new version enables one to study some variational problems on stratified subanalytic pseudomanifolds. We first achieve an isomorphism between this rectifiable currents’ version and the version using subanalytic chains. Then we define a suitably modified mass on the complex of rectifiable currents to ensure that each sequence of subanalytic chains with bounded modified mass has a convegent subsequence and the limit rectifiable current still satisfies the perversity condition of the approximating chains. The associated mass minimizers turn out to be almost minimal currents and this fact leads to some regularity results.

Keywords

Variational Problem Regularity Result Modify Mass Mass Minimizer Homology Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Bombieri: Regularity theory for almost Minimal currents, p. 99-130.Google Scholar
  2. 2.
    A. Borel: Intersection cohomology, Progress in mathematics 50. Birkhäuser Boston, Basel 1984.Google Scholar
  3. 3.
    H. Federer: Geometric Measure Theory. Springer-Verlag, New York, 1969.Google Scholar
  4. 4.
    H. Federer: Dimension and measure. Trans. Amer. Math. Soc. 62 (1947), 536-547.Google Scholar
  5. 5.
    M. Goresky, R. MacPherson: Intersection homology theory. Topology 19 (1980), 135-162.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Goresky, R. MacPherson: Intersection homology theory II. Inv. Math. 72 (1983), 77-130.MathSciNetzbMATHGoogle Scholar
  7. 7.
    R. Hardt, L. Simon: Seminar on Geometric Measure Theory. Birkhäuser, Boston, 1986.Google Scholar
  8. 8.
    R. Hardt, D. Kinderlehrer: Some regularity results in ferromagnetism. Comm. Partial Differential Equations 25 (2000), 1235-1258.MathSciNetzbMATHGoogle Scholar
  9. 9.
    R. Hardt: Topological properties of subanalytic sets. Trans. Amer. Math. Soc. 211 (1975), 57-70.Google Scholar
  10. 10.
    R. Hardt: Triangulation of subanalytic sets and proper light subanalytic maps. Invent. Math. 38 (1976/77), 207-217.Google Scholar
  11. 11.
    H. Hironaka: Introduction to Real-Analytic sets and real-analytic maps. Istituto Matematico “L.Tonelli” Dell’ Universita’DI Pisa, 1973.Google Scholar
  12. 12.
    F. Kirwan: An introduction to intersection homology theory. Longman Scientific & Technical, 1988.Google Scholar
  13. 13.
    M. Li; Currents on spaces with cone-like singularities. Ph.D thesis, Rice University, 1995.Google Scholar
  14. 14.
    F. Morgan: Geometric Measure theory, a beginner’s guide. Academic Press, 1995.Google Scholar
  15. 15.
    L.A. Santalo: Integral Geometry and Geometric Probability. Encyclopedia of mathematics and its applications, Volume I 1976.Google Scholar
  16. 16.
    L. Simon: Lectures on geometric measure theory. Proc. Centre Math. Anal. Australian National University 3 (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • Qinglan Xia
    • 1
  1. 1.Department of MathematicsRice UniversityUSA

Personalised recommendations