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 sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler-Lagrange equations. The resulting mean curvature operator can be of third order.

arXiv commentFinal version accepted in Calc. Var. Partial Differential Equations

References in corpus (3)

• Existence of isoperimetric regions in contact sub-Riemannian manifolds
• First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds
• Submanifolds and differential forms on Carnot manifolds, after M. Gromov and M. Rumin

Cited by in corpus (2)

• Regularity of Lipschitz boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group ${\mathbb{H}}^1$
• Higher dimensional holonomy map for ruled submanifolds in graded manifolds