Variational formulas for submanifolds of fixed degree
Abstract
We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in subRiemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the EulerLagrange equations. The resulting mean curvature operator can be of third order.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.05131
 Bibcode:
 2019arXiv190505131C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Metric Geometry;
 49Q05;
 53C42;
 53C17