A class of complete minimal submanifolds and their associated families of genuine deformations

摘要

Concerning the problem of classifying complete submanifolds of Euclideanspace with codimension two admitting genuine isometric deformations, until nowthe only known examples with the maximal possible rank four are the realKaehler minimal submanifolds classified by Dajczer-Gromoll \cite{dg3} inparametric form. These submanifolds behave like minimal surfaces, namely, ifsimple connected either they admit a nontrivial one-parameter associated familyof isometric deformations or are holomorphic. In this paper, we characterize a new class of complete minimal genuinelydeformable Euclidean submanifolds of rank four but now the structure of theirsecond fundamental and the way it gets modified while deforming is quite moreinvolved than in the Kaehler case. This can be seen as a strong indication thatthe above classification problem is quite challenging. Being minimal, thesubmanifolds we introduced are also interesting by themselves. In particular,because associated to any complete holomorphic curve in $\C^N$ there is such asubmanifold and, beside, the manifold is not Kaehler.