Partial regularity of minimizers for real-analytic sub-Riemannian metrics

TL;DR

This paper proves that in real-analytic sub-Riemannian geometry, minimizers are mostly real-analytic except on a small, countable set, and provides conditions to eliminate non-analytic minimizers, enhancing understanding of their regularity.

Contribution

It establishes partial regularity of sub-Riemannian minimizers and introduces a geometric condition to exclude strictly abnormal minimizers, ensuring full analyticity.

Findings

Minimizers are real-analytic except on a countable, non-dense set.

Non-analyticity occurs only on measure-zero sets.

A geometric condition guarantees the absence of strictly abnormal minimizers.

Abstract

In the real-analytic setting, we show that all sub-Riemannian minimizers (parametrized by the arc-length) are real-analytic everywhere except an at most countable non-dense set. In particular, non-analyticity may occur only on a set of measure zero of the domain of definition of a sub-Riemannian minimizer. Furthermore, we provide a geometrical condition which implies the absence of the so called strictly abnormal minimizers. In particular, under such condition, all sub-Riemannian minimizers (parametrized by the arc-length) are real-analytic.