Variations for submanifolds of fixed degree

Abstract

The aim of this PhD thesis is to study the area functional for submanifolds immersed in an equiregular graded manifold. This setting extends the sub-Riemannian one, removing the bracket generating condition. However, even in the sub-Riemannian setting only submanifolds of dimension or codimension one have been extensively studied. We will study the general case and observe that in higher codimension new phenomena arise, which do not show up in the Riemannian case. In particular, we will prove the existence of isolated surfaces, which do not admit degree preserving variations: a phenomena observed up to now only for curves, related to the notion of abnormal geodesics.