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