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A note on the diameter of small sub-Riemannian balls

Marco Di Marco, Gianluca Somma, Davide Vittone

Abstract

We observe that the diameter of small (in a locally uniform sense) balls in $C^{1,1}$ sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to $C^0$, the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.

A note on the diameter of small sub-Riemannian balls

Abstract

We observe that the diameter of small (in a locally uniform sense) balls in sub-Riemannian manifolds equals twice the radius. We also prove that, when the regularity of the structure is further lowered to , the diameter is arbitrarily close to twice the radius. Both results hold independently of the bracket-generating condition.
Paper Structure (4 sections, 4 theorems, 27 equations)

This paper contains 4 sections, 4 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.1
  3. Proof of Theorem 1.2
  4. Existence of calibrations

Key Result

Theorem 1.1

Let $M$ be a smooth manifold endowed with a $C^{1,1}$ sub-Riemannian structure. Then, for every $p \in M$ there exist a neighbourhood $V$ of $p$ and $r_p>0$ such that

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof : Proof of Theorem \ref{['teo_liscio']}
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • proof : Proof of Theorem \ref{['eps diam']}
  • ...and 3 more