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

Abstract

Concerning the problem of classifying complete submanifolds of Euclidean space with codimension two admitting genuine isometric deformations, we believe that the only known examples with the maximal possible rank four are the real Kaehler minimal submanifolds classified in parametric form by Dajczer-Gromoll [11]. They behave like minimal surfaces, namely, if simple connected they are either holomorphic or they admit a nontrivial one-parameter family of isometric deformations. Here we characterize a new class of complete genuinely deformable Euclidean submanifolds of rank four but now the structure of their second fundamental and the way it gets modified while deforming is quite more involved than in the Kaehler case. In particular, we see this as a strong indication that the classification problem is quite challenging. Being minimal, the submanifolds we introduced are also interesting by themselves. Some of the very basic question in the local and global theory of isometric immersions of Riemannian manifolds into Euclidean space remain in good part unanswered. For instance, outside some special cases it is not known which is the lowest codimension for which a given Riemannian manifold admits an isometric immersion. On one hand, there are several results that assure that a submanifold must be unique, that is, isometrically rigid, when lying in its lowest possible codimension. On the other hand, there are only a few theorems classifying isometrically deformable submanifolds and their deformations. This is due to the fact that rigidity is a “generic” property while being deformable is certainly not, and hence a situation harder to deal with. The exception for the deformation problem is the case of hypersurfaces. In fact, in the local case the problem was mostly solved by Sbrana [18] and Cartan [1] about a century ago; see [7] for details and a modern presentation. A solution to the problem for compact hypersurfaces was given by Sacksteder [17] and by Dajczer-Gromoll [10] in the complete case. But solving the deformation problem in codimension two turned out to be very challenging even in the more restrictive case of complete manifolds. In dealing with the isometric deformation problem in higher codimension, it has to be taken into account that any submanifold of a deformable submanifold has the isometric