Area-stationary surfaces inside the sub-Riemannian three-sphere

Abstract

ABSTRACT. We consider the sub-Riemannian metric g h on S 3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus.We study area-stationary surfaces with or without a volume constraint in (S 3 , g h ). By following the ideas and techniques in [RR2] we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volumepreserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot-Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C 2 compact, connected, embedded surfaces in (S 3 , g h ) with empty singular set and constant mean curvature H such that H/ √ 1 + H 2 is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (S 3 , g h ).

Sub-Riemannian Geodesics on the 3-D Sphere

“…This is a Lagrangian approach. The Lagrangian formalism was applied to study the sub-Riemannian geometry of S 3 in [1,3]. In the Riemannian geometry the minimizing curve locally coinsides with the geodesic, but it is not the case for the sub-Riemannian manifolds.…”

Modified Action and Differential Operators on the 3-D Sub-Riemannian Sphere

“…The missing direction is also can be obtained as an integral line of the Hopf vector field corresponding to the Hopf fibration. The sub-Riemannian geometry on S 3 was studied in [17,19,25], see also [14]. Explicit formulas for geodesics were given in [19].…”

“…Let us mention that the word 'geodesic' in our terminology stands for the projection of the solutions to a Hamiltonian system onto the underlying manifold, that is a good generalization of the notion of geodesic from Riemannian to sub-Riemannian manifolds, see for instance [33,38]. The Lagrangian approach was applied in [17] and [25] in order to characterize and to find the shortest geodesics. Another approach based on the control theory was employed in [14].…”

Sub-Riemannian geometry of parallelizable spheres

“…[2,6,9,14,17]. Due to its algebraic structure, it is sufficient to define appropriate distributions at the identity of the group.…”

“…Namely, in [17] it is introduced the mapping J(Z) = ∇ Z V , for Z ∈ T S 3 , were ∇ denotes the Levi-Civita connection on the tangent bundle to S 3 and V is the vector field defined in Section 2.…”