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.

Answers:

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 ?

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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