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.

Variational formulas for submanifolds of fixed degree / Citti, G; Giovannardi, G; Ritore, M. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - ELETTRONICO. - 60:233:(2021), pp. 1-44. [10.1007/s00526-021-02100-8]

### 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.
##### Scheda breve Scheda completa Scheda completa (DC) 2021
60:233
1
44
Citti, G; Giovannardi, G; Ritore, M
File in questo prodotto:
File
s00526-021-02100-8.pdf

Accesso chiuso

Tipologia: Pdf editoriale (Version of record)
Licenza: Open Access
Dimensione 623.52 kB
Utilizza questo identificatore per citare o creare un link a questa risorsa: `https://hdl.handle.net/2158/1284587`
• ND
• 3
• 1