On complete developable submanifolds in complex Euclidean spaces
The authors study immersed, complete, complex submanifolds \(M^n\) in \({\mathbb C}^N\) which they call developable, which means that the image of such a submanifold's Gauss map \(\Gamma\colon M^n \to G_{\mathbb C}(n,N)\) into the complex Grassmannian has dimension \(r<n\). Recall that \(\Gamma(x)\) is the subspace of \({\mathbb C}^N\) parallel to the tangent space \(T_x(M^n)\). The level sets of \(\Gamma\) are \((n-r)\)-dimensional linear subvarieties of \({\mathbb C}^N\). If \(r=1\) they are all parallel to each other, and hence \(M^n\) is a cylinder. This is the complex analogue due to Abe of the classical Hartman-Nirenberg theorem. When \(r\geq 2\), there are non-cylinder examples found by Dajczer-Gromoll, Bourgain, Wu and Vitter. In this interesting paper, the authors first prove that when \(r=2\) and \(M^n\) is not a cylinder, then \(M^n\) is the total space of a holomorphic fiber bundle over a Riemann surface. The second main result concerns the general case when the rank restriction is removed. In the last part of paper, the authors raise some questions related to topological aspects of developable submanifolds and the holomorphic deformability of a developable submanifold into a cylinder.
scientific article; zbMATH DE number 1893858