Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds

Tommaso Rossi, Alec Jacopo Almo Schiavoni Piazza, Alessandro Socionovo

Abstract

We study the regularity and branching of strictly abnormal minimizing geodesics in sub-Riemannian geometry. We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions.

Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds

Abstract

We study the regularity and branching of strictly abnormal minimizing geodesics in sub-Riemannian geometry. We construct examples of real-analytic sub-Riemannian manifolds admitting minimizing geodesics that lose regularity at an interior point of their domain and exhibit branching, thereby resolving longstanding open questions.
Paper Structure (19 sections, 34 theorems, 198 equations)

This paper contains 19 sections, 34 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. The regularity problem
  3. The branching problem
  4. The main result
  5. Further directions
  6. Structure of the paper
  7. Preliminaries and reduction to R2
  8. A new Liu--Sussmann Lemma
  9. Length estimates on the level sets of P̃
  10. Geometric obstructions to optimality
  11. Winding number of plane curves and weighted area
  12. Geometry of plane curves
  13. Comparison of curves through cut-and-paste arguments
  14. Loops of optimal competitors
  15. Uniqueness of the loop
  16. ...and 4 more sections

Key Result

Theorem 1.1

There exist $\varepsilon,s>0$ such that the curves $\gamma_{s,\varepsilon},\bar{\gamma}_{s,\varepsilon}$ are length-minimizing in $\mathcal{M}$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 64 more