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Annals of Global Analysis and Geometry

, Volume 26, Issue 3, pp 253–269 | Cite as

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

  • M. Carmen Domingo-Juan
  • Vicente Miquel
Article

Abstract

Let D be a domain obtained by a holomorphic motion of a domain Dp⊂ ℂ M λp n−1 along a complex curve P in a complex space form ℂ M λ n . We prove that, if n= 2, the volume of D depends only on the geometry of Dp 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 Dp, a similar result holds only for ‘Frenet motions’, but when Dp has certain integral symmetries (and only in this case) this result is still true for any motion .

Pappus formulae tube volume complex space form holomorphic motion along a complex submanifold 

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© Kluwer Academic Publishers 2004

Authors and Affiliations

  • M. Carmen Domingo-Juan
  • Vicente Miquel

There are no affiliations available

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