[Paper Review] On the regularity of abnormal minimizers for rank $2$ sub-Riemannian structures

Davide Barilari, Yacine Chitour+3 more|arXiv (Cornell University)|Apr 3, 2018

Analytic and geometric function theory29 references40 citations

TL;DR

Proves C1 regularity for a class of abnormal length-minimizers in rank-2 sub-Riemannian structures; deduces C1 for all such minimizers when step ≤ 4, with further smoothness under nilpotency conditions.

ABSTRACT

We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.

Motivation & Objective

• Motivate the regularity problem for length-minimizers in sub-Riemannian geometry and identify the gap for strictly abnormal curves.
• Establish conditions under which abnormal minimizers in rank-2 structures achieve C1 regularity.
• Show that in rank-2 structures of step up to 4, all length-minimizers are C1.
• Provide a pathway to higher regularity (C∞) under nilpotency assumptions on the generating Lie algebra.
• Connect the dynamic (Hamiltonian) structure of abnormal extremals to geometric regularity results.

Proposed method

• Study abnormal extremals via their Hamiltonian dynamics and the associated dual variables h_i and h_{i1...im}.
• Desingularize and nilpotentize the problem to reduce to equiregular, Carnot-like models.
• Derive and analyze a 2x2 system for h=(-h_{212}, h_{112}) evolving with u in S^1, using matrix A(t) with zero trace.
• Prove that on intervals where the relevant function h ≠ 0, the optimal control aligns with eigenvectors of A, and use time-rescaling to study limits at potential singular points.
• Use a non-positivity result for det A at interval endpoints to exclude corner-type minimizers under certain conditions.

Experimental results

Research questions

• RQ1 Does every length-minimizer in rank-2 sub-Riemannian structures admit C1 regularity when abnormal lifts avoid (D^4)⊥?
• RQ2 Under what structural assumptions (e.g., step ≤ 4) do all length-minimizers become C1 in rank-2 sub-Riemannian manifolds?
• RQ3 What additional conditions lead to C∞ regularity of length-minimizers beyond C1?
• RQ4 How does desingularization and nilpotent approximation influence regularity of abnormal minimizers?
• RQ5 Can corners be excluded as length-minimizers via Hamiltonian dynamics for rank-2, step-bounded cases?

Key findings

• If an abnormal minimizer in a rank-2 structure has a lift with λ(t) ∉ (D^4)⊥ for all t, then it is C1.
• A corollary: when the rank-2 sub-Riemannian structure has step ≤ 4, all length-minimizers are C1.
• Under a nilpotent generation condition (Lie{X1,X2} of step ≤ 4), length-minimizers are actually C∞.
• For the general case, the dynamics of abnormal extremals yields a dichotomy where non-smoothness would imply corner-like singularities, which can be ruled out under the stated hypotheses.
• The analysis relies on the Goh conditions (h1 = h2 = h12 = 0) and the evolution dh/dt = u1 h1? + u2 h2? with A(t) having zero trace.
• Desingularisation reduces non-equiregular cases to equiregular models, preserving the core regularity conclusions.