## Notes of Frédéric Jean’s lecture

A class of ${(H^k,1)}$-rectifiable subsets of metric spaces (and Hausdorff measures of curves in SR geometry)

Joint work with R. Ghezzi.

1. Case of ${C^1}$ curves

1.1. ${k}$-length

Let distribution ${D^1}$ define a sub-Riemannian structure on ${M}$. Let ${D^1 \subset D^2 \subset\cdots}$ denote the distributions generated by brackets. If ${\gamma}$ is a ${C^1}$ curve tangent to ${D^k}$, then ${dim_{Haus}(\gamma)=k}$. We give a formula for Hausdorff measure. When ${k=1}$, ${\mathcal{H}^1}$ equals length.

With Falbel, I defined a few years ago the metric differential of a curve,

$\displaystyle \begin{array}{rcl} meas_t^k(\gamma)=\limsup_{s\rightarrow t}\frac{d(\gamma(t),\gamma(t))}{s^{1/k}}, \end{array}$

which allows to define

$\displaystyle \begin{array}{rcl} \mathrm{Length}^k(\gamma)=\int meas_t^k(\gamma)\,dt. \end{array}$

Question. In what generality does

$\displaystyle \begin{array}{rcl} \mathcal{H}^k (\gamma)=\mathrm{Length}^k(\gamma) \end{array}$

hold ?

1.2. Complexity

An ${\epsilon}$-chain over a curve ${\gamma}$ is an ordered set of points ${q_1,\ldots,q_k}$ with ${d(q_i,q_{i+1})<\epsilon}$.

Definition 1 Let ${\gamma\in M}$ be a curve. Let ${\sigma(\gamma,\epsilon)}$ be the minimal number of points in an ${\epsilon}$-chain of ${\gamma}$. The complexity of ${\gamma}$ is

$\displaystyle \begin{array}{rcl} \sigma^k (\gamma)=\lim_{\epsilon\rightarrow 0}\epsilon^{k}\sigma(\gamma,\epsilon). \end{array}$

Question. Compare complexity and Hausdorff measure.

Falbel-Jean: for ${k=2}$ in contact and quasi-contact case.

Gauthier-Zakalyukin: for co-rank ${1}$ and ${2}$ distributios, and a few other cases.

1.3. Results

Say a curve is equiregular if ${D^1,\ldots,D^k}$ have constant rank along ${\gamma}$.

Theorem 2 (Falbel-Jean) Let ${\gamma}$ be a ${C^1}$ embedded curve in ${M}$. Assume that ${\gamma}$ is tangent to ${D^k}$. Assume that ${\gamma}$ is equiregular. Then

$\displaystyle \begin{array}{rcl} \mathcal{H}^k (\gamma)=\mathrm{Length}^k(\gamma)=\sigma^k(\gamma). \end{array}$

Furthermore, if ${\dot{\gamma}(t)\notin D^{k-1}(q)}$, ${q=\gamma(t)}$, then

$\displaystyle \begin{array}{rcl} \lim_{r\rightarrow 0}\frac{\mathcal{H}^k (\gamma)\cap B(q,r)}{2.r^k}=1. \end{array}$

2. Rectifiability

2.1. Euclidean case

In Euclidean space, a set ${A}$ is ${(\mathcal{H}^k,k)}$-rectifiable if ${\mathcal{H}^k}$-almost all of it is the union of a countable collection of Lipschitz images of ${{\mathbb R}^k}$. One may replace Lipschitz by ${C^1}$.

Theorem 3 (Federer, Preiss) Let ${A}$ have finite ${\mathcal{H}^k}$ measure. Then ${A}$ is ${(\mathcal{H}^k,k)}$-rectifiable if and only if

$\displaystyle \begin{array}{rcl} \lim_{r\rightarrow 0}\frac{\mathcal{H}^k (A\cap B(q,r))}{r^k} \end{array}$

exists and is ${\mathcal{H}^k}$-almost everywhere constant.

Rectifiability implies a.e. differentiability (existence of a tangent ${k}$-plane).

2.2. ${m-C_k^1}$-curves

The situation is pretty different in sub-Riemannian geometry. Let us concentrate on curves now. Lipschitz images of ${{\mathbb R}}$ are horizontal curves, so ${(\mathcal{H}^k,k)}$-rectifiable curves of positive measure exist only if ${k=1}$ and they are horizontal.

This forces us to enlarge the class of model curves to be used in the definition of rectifiability.

Definition 4 Let ${X}$ be a metric space. Let ${\gamma}$ be a continuous curve in ${X}$. Say that ${\gamma}$ admits a metric derivative of degree ${k}$ at ${t}$ if the limit

$\displaystyle \begin{array}{rcl} meas_t^k (\gamma)=\lim_{s\rightarrow 0}\frac{d(\gamma(t),\gamma(t+s))^k}{|s|} \end{array}$

exists everywhere. Say ${\gamma}$ is metrically ${C^1}$ of degree ${k}$ (abbreviated into ${m-C_k^1}$) if ${t\mapsto meas_t^k(\gamma)}$ is continuous.

Such curves are ${\frac{1}{k}}$-Hölder continuous, locally bi-Hölder at points where ${meas_t^k (\gamma)>0}$.

Proposition 5 1. In Euclidean (or Riemannian) geometry, ${m-C_k^1}$-curves with positive ${meas_t^k}$ are Lipschitz and ${k=1}$.

2. Let ${\gamma\subset M}$ be a ${C^1}$ and equiregular curve is a sub-Riemannian manifold. Let ${k}$ be the smallest integer such that ${\dot{\gamma}\subset D^k}$. Then ${\gamma}$ is ${m-C_k^1}$ and ${meas_t^k(\Gamma)}$ does not vanish everywhere.

Remark 1 Assume ${\gamma\subset M}$ is absolutely continuous (in Riemannian sense) and ${m-C_k^1}$, then ${k}$ is the smallest integer such that ${\dot{\gamma}\subset D^k}$.

It turns out that Theorem 1 extends to arbitrary metric spaces.

Theorem 6 (Ghezzi-Jean) Let ${X}$ be a metric space. Let ${\gamma}$ be an ${m-C_k^1}$ injective curve in ${X}$. Then

$\displaystyle \begin{array}{rcl} \mathcal{H}^k (\gamma)=\mathrm{Length}^k(\gamma)=\sigma^k(\gamma). \end{array}$

Furthermore, if ${meas_t^k(\gamma)>0}$, then density exists at at ${q=\gamma(t)}$, i.e.

$\displaystyle \begin{array}{rcl} \lim_{r\rightarrow 0}\frac{\mathcal{H}^k (\gamma)\cap B(q,r)}{2.r^k}=1. \end{array}$

I expect that ${m-C_k^1}$-curves in sub-Riemannian manifolds even have a derivative (an element in the tangent Carnot group), not merely a numerical metric differential.

2.3. Rectifiability of curves in metric spaces

Definition 7 Let ${X}$ be a metric space. A set ${A}$ is ${(\mathcal{H}^k,k)}$-rectifiable if ${\mathcal{H}^k}$-almost all of it is the union of a countable collection of images of ${m-C_k^1}$ curves.

Theorem 8 (Ghezzi-Jean) Let ${X}$ be a metric space. Let ${A}$ be a ${(\mathcal{H}^k,k)}$-rectifiable set with finite ${\mathcal{H}^k}$-measure. Then

$\displaystyle \begin{array}{rcl} 1\leq\liminf_{r\rightarrow 0}\frac{\mathcal{H}^k (\gamma)\cap B(q,r)}{2.r^k}\leq \limsup_{r\rightarrow 0}\frac{\mathcal{H}^k (\gamma)\cap B(q,r)}{2.r^k}\leq 2^k. \end{array}$

I am convinced that density exists and is equal to 2.

${m-C_k^1}$ is not preserved by bi-Lipschitz homeomorphisms, so our definition of ${(\mathcal{H}^k,k)}$-rectifiability is not bi-Lipschitz invariant.

Question. Can one cook up a bi-Lipschitz invariant definition and have the same conclusion ?