Existence, multiplicity, and regularity for sub-Riemannian geodesics by variational methods

A variational theory for geodesics joining a point and a one dimensional submanifold of a sub-Riemannian manifold is developed. Given a Riemannian manifold (M, g), a smooth distribution Δ ⊂ TM of codimension one in M, a point p ∈ M, and a smooth immersion γ :R → M with closed image in M and which is everywhere transversal to Δ, we look for curves in M that are stationary with respect to the Riemannian energy functional among all of the absolutely continuous curves horizontal with respect to Δ and that join p and γ. If (M,g) is complete, such extremizers exist, and they are curves of class C² characterized as the solutions of an integro-differential equation or by a system of ordinary differential equations. We also present some results concerning a sort of exponential map relative to the integro-differential equation and some applications. In particular, we obtain that if p and γ are sufficiently close in M, then there exists a unique length minimizer. We obtain existence and multiplicity results by means of the Ljusternik-Schnirelman theory.