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. - STAMPA. - 60:6(2021), pp. 233.1-233.44. [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.
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2021
Variational formulas for submanifolds of fixed degree / Citti G.; Giovannardi G.; Ritore M.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 60:6(2021), pp. 233.1-233.44. [10.1007/s00526-021-02100-8]
Citti G.; Giovannardi G.; Ritore M.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/880661`