Variational formulas for submanifolds of fixed degree
MetadataShow full item record
Sub-Riemannian manifoldsGraded manifoldsDegree of a submanifoldAdmissible variationsIsolated submanifolds
Citti, G., Giovannardi, G. & Ritoré, M. Variational formulas for submanifolds of fixed degree. Calc. Var. 60, 233 (2021). [https://doi.org/10.1007/s00526-021-02100-8]
SponsorshipHorizon 2020 Project ref. 777822: GHAIA; MEC-Feder grants MTM2017-84851-C2-1-P and PID2020-118180GB-I00; Junta de Andalucía grants A-FQM-441-UGR18 and P20-00164; Research Unit MNat SOMM17/6109 and PRIN 2015 “Variational and perturbative aspects of nonlinear differential problems”; Universidad de Granada/CBUA
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.