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The Analytic Minimal Rank Sard Conjecture

A Belotto da Silva, A Parusiński, L Rifford

TL;DR

The paper proves the minimal rank Sard Conjecture in the analytic category under a new qualitative hypothesis called splittability. It introduces witness transverse sections to obtain uniform, dimension-controlled reductions of the singular foliation and leverages the abnormal distribution vec{K} on Δ^{⊥} to lift singular horizontal curves and derive a contradiction when a positive-measure set of minimal-rank abnormal endpoints would exist. The framework generalizes prior three-dimensional results to arbitrary dimensions by replacing delicate singularity arguments with subanalytic and symplectic techniques, and it yields corank-1 corollaries (e.g., Sard holds for n=4, m=3) and insights into when Sard-type conclusions fail (via a non-splittable foliation example). The results advance the understanding of accessibility and Sard-type phenomena in high-dimensional sub-Riemannian geometry, while highlighting the role of splittability and witness sections in controlling singular behavior.

Abstract

We obtain, under an additional assumption on the subanalytic abnormal distribution constructed in [4], a proof of the minimal rank Sard conjecture in the analytic category. It establishes that from a given point the set of points accessible through singular horizontal curves of minimal rank, which corresponds to the rank of the distribution, has Lebesgue measure zero. The minimal rank Sard Conjecture is equivalent to the Sard Conjecture for co-rank 1 distributions.

The Analytic Minimal Rank Sard Conjecture

TL;DR

The paper proves the minimal rank Sard Conjecture in the analytic category under a new qualitative hypothesis called splittability. It introduces witness transverse sections to obtain uniform, dimension-controlled reductions of the singular foliation and leverages the abnormal distribution vec{K} on Δ^{⊥} to lift singular horizontal curves and derive a contradiction when a positive-measure set of minimal-rank abnormal endpoints would exist. The framework generalizes prior three-dimensional results to arbitrary dimensions by replacing delicate singularity arguments with subanalytic and symplectic techniques, and it yields corank-1 corollaries (e.g., Sard holds for n=4, m=3) and insights into when Sard-type conclusions fail (via a non-splittable foliation example). The results advance the understanding of accessibility and Sard-type phenomena in high-dimensional sub-Riemannian geometry, while highlighting the role of splittability and witness sections in controlling singular behavior.

Abstract

We obtain, under an additional assumption on the subanalytic abnormal distribution constructed in [4], a proof of the minimal rank Sard conjecture in the analytic category. It establishes that from a given point the set of points accessible through singular horizontal curves of minimal rank, which corresponds to the rank of the distribution, has Lebesgue measure zero. The minimal rank Sard Conjecture is equivalent to the Sard Conjecture for co-rank 1 distributions.
Paper Structure (12 sections, 15 theorems, 88 equations, 5 figures)

This paper contains 12 sections, 15 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Witness transverse sections
  4. Splittable foliation
  5. Paper structure
  6. Splittable foliations
  7. Sufficient conditions for splittability
  8. Example of a non-splittable foliation
  9. Witness transverse sections
  10. Proof of Theorem \ref{['thm:Witness']}.
  11. Proof of Theorem \ref{['thm:CorWitness']}.
  12. Proof of Theorem \ref{['thm:Sardminrank']}

Key Result

Theorem 1.1

Assume that both $M$ and $\Delta$ are real-analytic. If the foliation $\vec{\mathcal{K}}_{|S_0}$ is splittable, then the minimal rank Sard conjecture holds true.

Figures (5)

  • Figure 1: Geometrical illustration of $\mathcal{F}$.
  • Figure 2: Local foliation chart and transverse sections
  • Figure 3: A picture showing the sets $\vec{K}, \vec{H}, \vec{P}, \vec{Q}, I$ and $\bar{W}^1$
  • Figure 4: A picture to illustrate Lemma \ref{['lem:MetricProperties']}
  • Figure 5: The sets $\mathcal{O}_j^{c,0}, \mathcal{O}_j^{c,1}$ and $\mathcal{M}_j^{c}$

Theorems & Definitions (28)

  • Theorem 1.1: Minimal rank Sard Conjecture for splittable foliatons
  • Corollary 1.2
  • Definition 1.3: Splittable foliation
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Theorem 3.1
  • Theorem 3.2
  • ...and 18 more