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

TL;DR

This paper introduces a new class of complete minimal submanifolds in Euclidean space with codimension two that admit genuine isometric deformations, expanding the understanding beyond previously known Kaehler minimal examples.

Contribution

It characterizes a novel class of minimal submanifolds with complex deformation structures, not limited to Kaehler geometry, and relates them to holomorphic curves in complex space.

Findings

New class of minimal submanifolds with rank four

Submanifolds admit genuine isometric deformations

Connection to holomorphic curves in ^N

Abstract

Concerning the problem of classifying complete submanifolds of Euclidean space with codimension two admitting genuine isometric deformations, until now the only known examples with the maximal possible rank four are the real Kaehler minimal submanifolds classified by Dajczer-Gromoll \cite{dg3} in parametric form. These submanifolds behave like minimal surfaces, namely, if simple connected either they admit a nontrivial one-parameter associated family of isometric deformations or are holomorphic. In this paper, we characterize a new class of complete minimal 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. This can be seen as a strong indication that the above classification problem is quite challenging. Being minimal, the submanifolds we introduced are also interesting by themselves. In particular, because associated to any complete holomorphic curve in $\C^N$ there is such a submanifold and, beside, the manifold in general is not Kaehler.