Skip to main content
SpringerLink
Log in

On the Volume of a Domain Obtained by a Holomorphic Motion Along a Complex Curve

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Let D be a domain obtained by a holomorphic motion of a domain D p⊂ ℂ M n−1λp along a complex curve P in a complex space form ℂ M nλ . We prove that, if n= 2, the volume of D depends only on the geometry of D p and the intrinsic geometry of P, but not on the extrinsic geometry of P. When M is closed (compact without boundary), then the dependence on P is only through its topology. When n > 2, and for arbitrary domains D p, a similar result holds only for ‘Frenet motions’, but when D p has certain integral symmetries (and only in this case) this result is still true for any motion .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Domingo-Juan, M. C., Gual, X. and Miquel, V.: Pappus type theorems for hypersurfaces in a space form, Israel J. Math. 128(2002), 205–220.

    Google Scholar 

  2. Domingo-Juan, M. C. and Miquel, V.: Pappus type theorems for motions along a submanifold, Differential Geom. Appl. 21(2004), 229–251.

    Google Scholar 

  3. Flaherty, F. J.: The volume of a tube in complex projective space, Illinois J. Math. 16(1982), 627–638.

    Google Scholar 

  4. Flanders, H.: A further comment of of Pappus, American Math. Monthly 77(1970), 9655–9968.

    Google Scholar 

  5. Goodman, W. and Goodman, G.: Generalizations of the theorems of Pappus, Amer. Math. Monthly 76(1969), 355–366.

    Google Scholar 

  6. Gray, A.: An estimate for the volume of a tube about a complex hypersurface, Tensor (N.S.) 39(1982), 303–305.

    Google Scholar 

  7. Gray, A.: Volumes of tubes about Kähler submanifolds expressed in terms of Chern classes, J. Math. Soc. Japan 36(1984), 23–35.

    Google Scholar 

  8. Gray, A.: Volumes of tubes about complex submanifolds of complex projective space, Trans. Amer. Math. Soc. 291(1985), 437–449.

    Google Scholar 

  9. Gray, A.: Tubes, 2nd edn, Birkhäuser, Boston, 2003.

    Google Scholar 

  10. Gray, A. and Miquel, V.: On Pappus-Type Theorems on the Volume in Space Forms, Ann. Global Anal. Geom. 18(2000), 241–254.

    Google Scholar 

  11. Griffiths, P. A.: Complex differential and integral geometry and curvature integrals associated to singularities on complex analytic varieties, Duke Math. J. 45(1978), 427–472.

    Google Scholar 

  12. Griffiths, P. and Harris, J.: Principles of Algebraic Geometry,Wiley, New York, 1978.

    Google Scholar 

  13. Pursell, L. E.: More generalizations of a theorem of Pappus, American Math. Monthly 77(1970), 961–965.

    Google Scholar 

  14. Thomas, G. B. and Finney, R. L.: Calculus and Analytic Geometry, Addison Wesley, Reading, Massachusetts, 1984.

    Google Scholar 

  15. Weyl, H.: On the volume of tubes, Amer. J. Math. 61(1939), 461–472.

    Google Scholar 

Download references

Authors
  1. M. Carmen Domingo-Juan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicente Miquel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Domingo-Juan, M.C., Miquel, V. On the Volume of a Domain Obtained by a Holomorphic Motion Along a Complex Curve. Annals of Global Analysis and Geometry 26, 253–269 (2004). https://doi.org/10.1023/B:AGAG.0000042929.53208.b6

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:AGAG.0000042929.53208.b6

Search

Navigation