On the volume of a domain obtained by a holomorphic motion along a complex curve (Q1888176)

One defines the notion of holomorphic motion along a complex curve \(P\) in \(\mathbb{C} M^n_\lambda\) (= the simply connected complex space form of real dimension \(2n\) and holomorphic sectional curvature \(4\lambda\)) and studies the volume of the domain \(D\) obtained by holomorphic motion of a domain \(D_p\). A similar formula for the volume of \(D\) to that for motions along curves in \(M^n_\lambda\) is given. By using this formula one obtains: If \(n= 2\), then volume\((D)\) depends only on \(D_p\) and the intrinsic geometry of \(P\). If \(n> 2\), then for the Frénet motion, the volume\((D)\) does not depend on normal curvatures and for not Frénet motions, the volume\((D)\) depends on the motion. The volume\((D)\) does not depend on the motion if and only if \(D_p\) has some special integral symmetries.