Existence, Multiplicity, and Regularity for sub-Riemannian Geodesics by Variational Methods

Abstract

We develop a variational theory for geodesics joining a point and a one dimensional submanifold of a sub-Riemannian manifold. Given a Riemannian manifold (M,g), a smooth distribution $\Delta\subset T{M}$ of codimension one in M, a point $p\in{M}$, and a smooth immersion $\gamma:\R\rightarrow{M}$ with closed image in M and which is everywhere transversal to $\Delta$, 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 $\Delta$ and that join p and $\gamma$. 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 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 $gamma$ 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.