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\title{Differential forms and the Hölder equivalence problem}
\author{P. Pansu}
\date{September 1st, 2014}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{prop}[lem]{Proposition}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{defi}[lem]{Definition}
\newtheorem{rem}[lem]{Remark}
\newtheorem{nota}[lem]{Notation}
\newtheorem{ques}[lem]{Question}
\newtheorem{prob}[lem]{Problem}
\newtheorem{sol}[lem]{Solution}
\newtheorem{conc}[lem]{Conclusion}
\newtheorem{conj}[lem]{Conjecture}
\newtheorem{ex}[lem]{Example}
\newtheorem{exs}[lem]{Examples}
\newtheorem{res}[lem]{Results}
\newtheorem{plan}[lem]{Plan of lecture}
\newtheorem{scheme}[lem]{Scheme of proof}

\def\proof{\par\medskip\noindent {\bf Proof}~}
\def\qed{~q.e.d.}

\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\Z{\mathbb{Z}}
\def\H{\mathbb{H}}
\def\O{\mathbb{O}}
\def\Q{\mathbb{Q}}
\def\n#1{{|\hskip-1pt | #1 |\hskip-1pt |}}
\def\eps{\epsilon}
\def\red#1{\textcolor{red}{#1}}

\mode<presentation>{\usetheme{Warsaw}}

\begin{document}


\begin{frame}
\titlepage
\end{frame}

\begin{frame}
$Heis$ denotes \emph{Heisenberg group}, the group of matrices of the form $\begin{pmatrix}
1&x&z    \\
0&1&y \\
0&0&1
\end{pmatrix}$ with real entries. \emph{Dilatations} $\delta_t(x,y,z)=(tx,ty,t^2 z)$ are group automorphisms. Its Lie algebra has basis
\begin{eqnarray*}
X=\begin{pmatrix}
1&x&z    \\
0&1&y \\
0&0&1
\end{pmatrix},\quad Y=\begin{pmatrix}
1&x&z    \\
0&1&y \\
0&0&1
\end{pmatrix},\quad Z=\begin{pmatrix}
1&x&z    \\
0&1&y \\
0&0&1
\end{pmatrix},
\end{eqnarray*}
with $[X,Y]=Z$ and all other brackets vanish. So $\{X,Y\}$ is a bracket generating. Its trajectories (paths tangent to $V_1=\mathrm{span}(\{X,Y\})$) are called \emph{horizontal paths}. The sub-Riemannian (or Carnot-Carath\'eodory) metric on $Heis$,
\begin{eqnarray*}
d(p,q)=\inf\{\mathrm{length}(\gamma)\,;\,\gamma\textrm{ horizontal path joining }p\textrm{ to }q\},
\end{eqnarray*}
is left-invariant and multiplied by $t$ by $\delta_t$.

\medskip

The distribution of planes $V_1$ is the kernel of the left-invariant differential 1-form $\theta=dz-x\,dy$.

\end{frame}

\begin{frame}

Small $R$-balls are squeezed in the $z$ direction, they have volume $R^4$, 
\begin{center}
\includegraphics[width=2in]{squeeze.pdf}
\end{center}

Hausdorff dimension equals 4. Vertical lines have Hausdorff dimension 2. Only (rectifiable) horizontal curves have finite 1-dimensional Hausdorff measure.

\bigskip

Metricly very different from Euclidean 3-space. How much ?

\end{frame}

\section{Metric problems in sub-Riemannian geometry}

\begin{frame}

\begin{center}
Lecture 1
\end{center}
\begin{center}
\textbf{Metric problems in sub-Riemannian geometry}
\end{center}

What do sub-Riemannian manifolds say, from the metric viewpoint ? Here is a choice of problems which are open even in dimension 3.
\begin{enumerate}
  \item Bi-Lipschitz equivalence of contact manifolds
  \item Bi-Lipschitz embedding $Heis$ in $L^1$
  \item Bi-Lipschitz embedding snowflakes of $Heis$ in $\R^N$
  \item H\"older equivalence problem
  \item H\"older-Lipschitz equivalence problem
  \item Quasi-symmetric H\"older-Lipschitz equivalence problem
\end{enumerate}
I comment on each of them. The sequel of the course will focus on 4.

\end{frame}

\subsection{Bi-Lipschitz equivalence of contact manifolds}

\begin{frame}

A \emph{contact structure} on a 3-manifold is a step 2 plane distribution.

\begin{thm}[\red{Bennequin 1983, Eliashberg 1989, 1992}]
Up to $C^1$ diffeomorphisms, there are exactly 2 different contact structures on $\R^3$.
\end{thm}

\begin{center}
\includegraphics[width=1.5in]{330px-Overtwisted_contact_structure.png}
\end{center}\pause

\begin{ques}
Does there exist a locally bi-Lipschitz homeomorphism between the tight (standard) contact structure and the overtwisted (Lutz) contact structure ?
\end{ques}
At first sight, none of the classical contact invariants (overtwisted disks, Lutz tubes, Thurston-Bennequin numbers,...) seems to be preserved by bi-Lipschitz homeomorphism. 

\end{frame}

\subsection{Bi-Lipschitz embedding $Heis$ in $L^1$}

\begin{frame}

\begin{thm}[\red{Semmes 1996, Pauls 2001}]
No bi-Lipschitz embedding of $Heis$ in $L^p$ if $1<p<\infty$.
\end{thm}

\proof A.e., there is a differential, a group homomorphism $Df:Heis\to L^p$, which is bi-Lipschitz. But $Df$ has a kernel, contradiction.\qed\pause

\medskip

Theoretical computer science (hardness of approximation of \textsf{SPARSEST CUT}) motivates the 

\begin{conj}[\red{Lee-Naor 2006}]
For every 1-Lipschitz map of the $R$-ball in $Heis$ into $L^1$, there are  points whose distance is decreased at least by a factor of $(\log R)^{1/2-\epsilon}$, $\forall\epsilon>0$.
\end{conj}\pause

Known \emph{for $L^1$ with some weaker exponent $\delta<\frac{1}{2}$} (\red{Cheeger-Kleiner-Naor 2010}). Quantitative differentiability.

Known \emph{if $L^1$ is replaced by $L^p$, $1<p\leq 2$, with sharp exponent $\frac{1}{2}$} (\red{Lafforgue-Naor 2012}). Functional inequality.

\medskip

Embedding theory raises new questions: $Heis^{m}$ is Markov 4-convex (Li, 2014).

\end{frame}

\subsection{Bi-Lipschitz embedding snowflakes of $Heis$ in $\R^N$}

\begin{frame}

\textbf{Remark}: a log perturbation of the Carnot metric bi-Lipschitz embeds in $L^2$. Can one do this in finite dimension ?

A \emph{snowflake} of metric space $(X,d_X)$ is $X^{1-\epsilon}=(X,d_X^{1-\epsilon})$.\hskip1em\includegraphics[width=1.2in]{snowflake.pdf}

\smallskip

The \emph{Assouad-Bouligand dimension} $\mathrm{dim}_{AB}(X)$ is the infimal $\delta$ such that the number of disjoint $r$-balls in an $R$-ball is $O((\frac{R}{r})^{\delta})$.

\begin{thm}[\red{Assouad 1983}]
If $\mathrm{dim}_{AB}(X)\leq\delta$, $X^{1-\epsilon}$ has a $L$-bi-Lipschitz embedding into $\R^N$, with $N=N(\delta,\epsilon)$, $L=L(\delta,\epsilon)$.
\end{thm}

Improved by \red{Naor-Neiman 2012}: $N=N(\delta)=O(\delta)$.\pause

\begin{ques}
What is the optimal embedding dimension for $Heis$ ?
\end{ques}

Easy: $N\geq 5$.

\end{frame}

\subsection{H\"older equivalence problem}

\begin{frame}

Remark: The obvious map $\R^3\to Heis$ is $C^{1/2}$-H\"older continuous, and its inverse is Lipschitz.

\begin{ques}[H\"older equivalence problem, \red{Gromov 1993}]
Let $M$ be a sub-Riemannian manifold. For which $\alpha\in(0,1)$ does there exist locally a homeomorphism from Euclidean space to $M$ which is $C^{\alpha}$-H\"older continuous ?
\end{ques}\pause

\begin{ques}[H\"older-Lipschitz equivalence problem]
Idem, but require a Lipschitz inverse.
\end{ques}\pause

\begin{defi}
Let $\alpha(M)=\sup\{\alpha\in(0,1)\,|\,\exists$ locally a $C^\alpha$ homeomorphism $\R^n \to M\}$.
\end{defi}\pause

\begin{thm}[\red{Gromov 1993}]
Let metric space $X$ have dimension $n$, Hausdorff dimension $Q$. Then $\alpha(X)\leq\frac{n}{Q}$.

Let sub-Riem. $M$ have dimension $n$, Hausdorff dimension $Q$. Then $\alpha(M)\leq\frac{n-1}{Q-1}$.

Let $M$ be a $2m+1$-dimensional contact manifold. Then $\alpha(M)\leq\frac{m+1}{m+2}$ ($\leq\frac{2m}{2m+1}$).
\end{thm}
 
\end{frame}

\subsection{Quasi-symmetric H\"older-Lipschitz equivalence problem}

\begin{frame}

\begin{defi}
Let $M$ be a Riemannian manifold. Let $-1\leq\delta<0$. Say $M$ is {\em $\delta$-pinched} if sectional curvature ranges between $-1$ and $\delta$. 
Define the {\em optimal pinching} $\delta(M)$ of $M$ as the least $\delta\geq-1$ such that $M$ is bi-Lipschitz to a $\delta$-pinched complete simply connected Riemannian manifold. \pause 
\end{defi}

Example: Complex hyperbolic plane $H^{2}_{\C}$ is $-\frac{1}{4}$-pinched. Is it true that $\delta(H^{2}_{\C})=-\frac{1}{4}$ ?\pause

\medskip

\textbf{Facts}. \begin{itemize}
  \item Negatively curved manifolds $M$ have a visual sphere $\partial M$, equipped with a visual metric.
  \item If $M$ is $\delta$-pinched, polar coordinates define a $C^{\sqrt{-\delta}}$-H\"older homeomorphism from the round sphere $S\to\partial M$, with 1-Lipschitz inverse.
  \item Bi-Lipschitz maps between negatively curved Riemannian manifolds induce {\em quasisymmetric} maps between ideal boundaries.
\end{itemize}

\begin{center}
\includegraphics[width=2in]{visual.pdf}\pause
\hskip1cm\includegraphics[width=1.5in]{qs.pdf}
\end{center}

\end{frame}

\begin{frame}

\begin{defi}
$\alpha_{qs}(X)=\sup\{\alpha\in(0,1)\,|\,\exists$ locally a $C^{\alpha}$ homeomorphism with Lipschitz inverse from Euclidean space to a metric space quasisymmetric to $X\}$
\end{defi}
By definition, $\sqrt{-\delta(M)}\leq \alpha_{qs}(\partial M)$.

\medskip

Example: The visual boundary of complex hyperbolic plane is a sub-Riemannian 3-sphere, quasisymmetric to $Heis$. Note that $\alpha_{qs}(Heis)\geq\alpha(Heis)\geq\frac{1}{2}$.\pause

\begin{conj}
$\alpha_{qs}(Heis)=\frac{1}{2}$.
\end{conj}

If the proof survives and time permits, I will explain 
\begin{thm}
$\alpha_{qs}(Heis)\leq\frac{2}{3}$.
\end{thm}

\end{frame}

\begin{frame}

\textbf{More on complex hyperbolic spaces}. The ball $B\subset\C^m$ has biholomorphism group $PU(m,1)$, which preserves a complete Riemannian metric on $B$ and the contact structure of complex tangents on $\partial B$. $PU(m,1)$ has a subgroup $S=\R\rtimes Heis^{2m-1}$ acting simply transitively on the ball. The conjugation action of $\R$ on $Heis^{2m-1}$ is by dilations. The induced metric on $S$ is of the form $dt^2+\delta_t^*g_0$, for some left-invariant Riemannian metric $g_0$ on $Heis^{2m-1}$. It is $-\frac{1}{4}$-pinched in the $t$ direction, since $\delta_t^*g_0=e^{2t}g_{V_1}+e^{4t}g_{V_2}$, thus in all directions by $PU(m,1)$-symmetry. $\R$ factors are geodesics, $Heis^{2m-1}$-orbits are horospheres. 

\begin{center}
\includegraphics[width=2in]{horosphere.pdf}
\end{center}

The visual sphere identifies with $\partial B$ equipped with a sub-Riemannian metric.

\end{frame}

\section{Gromov's dimension approach to the H\"older equivalence problem}

\begin{frame}
\begin{center}
Lecture 2
\end{center}
\begin{center}
\textbf{Gromov's dimension approach to the H\"older equivalence problem}
\end{center}

\bigskip

Source: Gromov's \emph{Carnot-Carath\'eodory spaces seen from within}.

\end{frame}

\subsection{Hypersurfaces}

\begin{frame}

Gromov uses Hausdorff dimension of subsets of given topological dimension: if all subsets of $X$ of topological dimension $k$ have Hausdorff dimension $\geq k'$, then $\alpha(X)\leq\frac{k}{k'}$.\pause

\begin{thm}[\red{Gromov, 1993}]
$G$ Carnot group of dimension $n$ and Hausdorff dimension $Q$. Then $\alpha(G)\leq\frac{n-1}{Q-1}$.
\end{thm}\pause

\textbf{Proof}. Use isoperimetric inequality for piecewise smooth domains $D\subset M$,
\begin{eqnarray*}
\mathrm{vol}(D)^{Q-1/Q}\leq const. \,\mathcal{H}^{Q-1}(\partial D).\quad(^*)
\end{eqnarray*}\pause

It follows that the boundary of any non smooth domain $\Omega$ has Hausdorff dimension at least $Q-1$. Indeed, cover $\partial \Omega$ with balls $B_j$ and apply (*) to $\Omega\cup\bigcup B_j$. This gives a lower bound on $\mathcal{H}^{Q-1}(\partial(\bigcup B_j))\leq\sum\mathcal{H}^{Q-1}(\partial B_j)\leq const.\,\sum\mathrm{diameter}(B_j)^{Q-1}$.

\begin{center}
\includegraphics[width=2in]{isop.pdf}
\end{center}

\end{frame}

\subsection{Euclidean isoperimetric inequality}

\begin{frame}

\proof of Euclidean isoperimetric inequality (with unsharp constant)
\begin{eqnarray*}
\mathrm{vol}(D)^{n-1/n}\leq const. \,\mathcal{H}^{n-1}(\partial D).
\end{eqnarray*}
Fix point $p$. Let $vol=dx_1\wedge ...\wedge dx_n$ be the volume form. Let $\xi(q)=\frac{q-p}{|q-p|^{n}}$ be a radial vectorfield and $\omega_p=\iota_{\xi}\omega$. \pause
Then $d\omega_p=0$ and the integral of $\omega$ on the boundary of every smooth domain containing $p$ equals 1. 
\pause
Use Fubini.
\begin{eqnarray*}
\mathrm{vol}(D)=\int_{D}(\int_{\partial D}\omega_p)\,dp\leq\int_{D\times\partial D}|p-q|^{1-n}\,dq\,dp=\int_{\partial D}(\int_{D}|p-q|^{n-1}\,dp)\,dq.
\end{eqnarray*}\pause
Replacing $D$ by a ball $B(q,R)$ with the same volume increases the integral,
\begin{eqnarray*}
\int_{D}|p-q|^{1-n}\,dp\leq\int_{B(q,R)}|p-q|^{1-n}\,dp=R=\mathrm{vol}(D)^{1/n}.
\end{eqnarray*}\pause
Key step is estimate $|\omega_p|\leq|p-q|^{1-n}$. It follows from homogeneity under dilations,
\begin{eqnarray*}
\delta_t^{*}\omega_0=\omega_0,
\end{eqnarray*}
since $n-1$-forms have pointwise homogeneity $n-1$.

\end{frame}

\subsection{Forms of weight $Q-1$}

\begin{frame}

On Carnot groups, $n-1$-forms come in several homogeneities called \emph{weights}.

\begin{defi}
On a Carnot group, a left invariant form $\lambda\in\Lambda^{k}\mathfrak{g}^*$ has \emph{weight} $w$ if $\delta_t^{*}\lambda=t^{w}\lambda$. A smooth differential form $\omega$ has weight $w$ if it is a linear combination of left-invariant forms of weight $w$.
\end{defi}

Example: on $Heis$, $dx$, $dy$, $\theta=dz-xdy$ are a basis of invariant 1-forms, with $dx$, $dy$ of weight 1 and $\theta$ of weight 2. $2xdx\wedge dy$ has weight 2, $(ydx+2xdy)\wedge\theta$ has weight 3.\pause

\begin{lem}
On a Carnot group $G$, if a smooth differential form $\omega$ of weight $w$ satisfies $\delta_2^{*}\omega=\omega$ on $G\setminus\{e\}$, then $|\omega(q)|\leq\mathrm{const.}\,d(q,e)^{-w}$.
\end{lem}
Indeed, $\{\delta_t^*\omega\,;\,t\in\R\}$ is a compact set, it is bounded on the unit sphere $S$. Let $\omega=\sum a_i \lambda_i$ with left-invariant $\lambda_i$. If $d(q,e)=t$, $q=\delta_t(q')\in S\subset G$,
\begin{eqnarray*}
d(q,e)^{w}|\omega|(q)=\sum |t^{w}a_i(q)|=|\sum a_i\circ\delta_t(q') \delta_t^*\lambda_i|=|\delta_t^*\omega|(q')\leq\mathrm{const.}
\end{eqnarray*}

\end{frame}

\begin{frame}

A closed $n-1$-form $\omega$ on $G\setminus\{e\}$ such that $\delta_2^{*}\omega_0=\omega_0$, with integral 1 on spheres, is a representative of the relevant class in $H^{n-1}(M)$, where $M=(G\setminus\{e\})/<\delta_2>$.\pause

\begin{lem}
Every cohomology class in $H^{n-1}(M)$ has a representative of weight $Q-1$.
\end{lem}

\proof for $Heis$. Let $\omega=a\,dx\wedge dy +\beta\wedge\theta$ be a closed 2-form. Then
\begin{eqnarray*}
\omega+d(a\,\theta)=(da+\beta)\wedge\theta
\end{eqnarray*}
has weight 3. \pause

\proof in general. All left-invariant $n-1$-forms are closed. Under dilations, the cohomology of left-invariant forms splits
\begin{eqnarray*}
H^{n-1}(\mathfrak{g})=\bigoplus^{w} H^{n-1,w}(\mathfrak{g}).
\end{eqnarray*}
Poincaré duality gives $H^{n-1,w}\simeq H^{1,Q-w}=0$ unless $Q-w=1$. Therefore for every left-invariant form $\lambda$ of weight $w\not=Q-1$, $\exists$ left-invariant $\mu$ of weight $w$ such that $d\mu=\lambda$.
Write $\omega=\sum a_i\lambda_i$, $\lambda_i$ left-invariant of weight $w_i$. If $\lambda_i=d\mu_i$, substracting $d(a_i\mu_i)$ replaces $a_i\lambda_i$ by $da_i\wedge\mu_i$ of weight $w_i+1$. After finitiely many steps, only terms of weight $Q-1$ remain.\qed


\end{frame}

\subsection{Hausdorff dimensions of higher codimensional subsets}

\begin{frame}

Recall the role of dimensions of subsets: if all subsets of topological dimension $k$ have Hausdorff dimension $\geq k'$, then $\alpha(M)\leq\frac{k}{k'}$.\pause

\medskip

To get lower bounds on Hausdorff dimension of subsets, Gromov constructs local foliations by horizontal submanifolds. If there are enough such dimension $k$ foliations, all subsets of topological dimension $n-k$ have Hausdorff dimension $\geq Q-k$, therefore $\alpha(M)\leq\frac{n-k}{Q-k}$.\pause

\medskip

Constructing horizontal submanifolds amounts to solving a system of PDE's. If $k=1$, it is an ODE, the method applies to all (equiregular) sub-Riemannian manifolds. Gromov solves the relevant PDE for contact $2m+1$-manifolds and $k=m$, and, more generally, for generic $h$-dimensional distributions, and $k$ such that $h-k\geq(n-h)k$. 

\end{frame}

\section{Gromov's cochain approach to the H\"older equivalence problem}

\begin{frame}
\begin{center}
Lecture 3
\end{center}
\begin{center}
\textbf{Gromov's cochain approach to the H\"older equivalence problem}
\end{center}

\bigskip

Source: Gromov's \emph{Carnot-Carath\'eodory spaces seen from within}.

\end{frame}

\subsection{Cochains}

\begin{frame}

\begin{defi}
On a metric space $X$, a (straight) {\em $q$-cochain of size} $\eps$ is a function $c$ on $q+1$-uples of diameter $\leq \eps$. Its $\eps$-absolute value is
\begin{eqnarray*}
|c|_{\eps}=\sup\{c(\Delta)\,;\,diam(\Delta)\leq \eps\}.
\end{eqnarray*} 
\end{defi}
In other words, straight cochains of size $\eps$ coincide with simplicial cochains on the simplicial complex whose vertices are points of $X$ and a $q$-face joins $q+1$ vertices as soon as all pairwise distances are $\leq \eps$. Therefore, they form a complex $\mathcal{C}_{\eps}^{.}$. There is a dual complex of chains $\mathcal{C}_{.,\eps}$. \pause

\begin{lemma}
Assume $X$ is a manifold with boundary, or bi-H\"older homeomorphic to such, then the inductive limit complex $\underrightarrow{\lim}$ $\mathcal{C}_{\eps}^{\cdot}$ computes cohomology. 
\end{lemma}

\begin{defi}
Given a cohomology class $\kappa$ and a number $\nu>0$, one can define the \emph{$\nu$-norm}
\begin{eqnarray*}
\n{\kappa}_{\nu}=\liminf_{\epsilon\to 0}\epsilon^{-\nu}\inf\{|c|_{\epsilon}\,|\,\textrm{ cochains } c \textrm{ of size } \epsilon \textrm{ representing } \kappa\}.
\end{eqnarray*}
\end{defi}

\end{frame}

\subsection{Metric weights}

\begin{frame}

\begin{defi}
Let $X$ be a metric space, let $q\in\mathbb{N}$. Define the {\em metric weight} $MW_q (X)$ as the supremum of numbers $\nu$ such that there exist arbitrarily small open sets $U\subset M$ and nonzero straight cohomology classes $\kappa\in H^q (U,\R)$ with finite $\nu$-norm $\n{\kappa}_{\nu}<+\infty$.
\end{defi}\pause

\begin{prop}
In Euclidean space, all straight cocycles $c$ representing a nonzero class $\kappa$ of degree $q$ satisfy
$|c|_{\epsilon}\geq \textrm{const.}(\kappa)\,\epsilon^q$.
In other words, $\n{\kappa}_{q}>0$.
\end{prop}

\proof.
Fix a cycle $c'$ such that $\kappa(c')>0$. Subdivide it as follows : fill simplices with affine singular simplices, subdivide them and keep only their vertices. This does not change the homology class. The number of simplices of size $\epsilon$ thus generated is $\leq \textrm{const.}(c')\,\epsilon^{-q}$. For any representative $c$ of size $\epsilon$ of $\kappa$,
\begin{eqnarray*}
\kappa(c')=c(c')\leq \textrm{const.}\,\epsilon^{-q} |c|_{\epsilon}.\qed
\end{eqnarray*}\pause

\begin{cor}
Euclidean $n$-space has $MW_q \leq q$ for all $q=1,\ldots,n-1$ (later, we shall see that $MW_q =q$).
\end{cor}

\end{frame}

\subsection{H\"older covariance}

\begin{frame}

\begin{prop}
\label{holderweight}
Let $f:X\to Y$ be a $C^{\alpha}$-H\"older continuous homeomorphism. Let $\kappa\in H^q (Y,\R)$. Then
\begin{eqnarray*}
\n{\kappa}_{\nu}<+\infty\Rightarrow\n{f^{*}\kappa}_{\nu\alpha}<+\infty.
\end{eqnarray*}
In particular, $MW_q (X)\geq \alpha MW_q (Y)$. 
Consequence: $\alpha(M)\leq\frac{q}{MW_q (M)}$ for all $q$.
\end{prop}

\proof.
If $\sigma$ is a straight simplex of size $\epsilon$ in $X$, $f(\sigma)$ has size $\epsilon' \leq \n{f}_{C^{\alpha}}\,\epsilon^{\alpha}$ in $Y$. If $c$ is a representative of $\kappa$, $f^* c$ is a representative of  $f^{*}\kappa$, and
\begin{eqnarray*}
\epsilon'^{-\nu}|c|_{\epsilon'}
&\geq&\epsilon'^{-\nu}|c(f(\sigma))|\\
&=&\epsilon'^{-\nu}|f^* c(\sigma)|\\
&\geq&\n{f}_{C^{\alpha}}^{-\nu}\,\epsilon^{-\nu\alpha}|f^* c(\sigma)|.
\end{eqnarray*}
Therefore
\begin{eqnarray*}
\epsilon^{-\nu\alpha}|f^* c|_{\epsilon}
\leq \n{f}_{C^{\alpha}}^{\nu}\,\epsilon'^{-\nu}|c|_{\epsilon'}.
\end{eqnarray*}
This leads to
\begin{eqnarray*}
\n{f^* \kappa}_{\nu\alpha}\leq \n{f}_{C^{\alpha}}^{\nu}\,\n{\kappa}_{\nu}.\qed
\end{eqnarray*}

\end{frame}

\subsection{Weights of differential forms}

\begin{frame}

Let $G$ be a Carnot group with Lie algebra $\mathfrak{g}$. Left-invariant differential forms on $G$ split into homogeneous components under the dilations $\delta_\epsilon$,
\begin{eqnarray*}
\Lambda^\cdot \mathfrak{g}^* =\bigoplus_{w}\Lambda^{\cdot,w} \quad \textrm{where}\quad \Lambda^{\cdot,w} =\{\alpha\,|\,\delta_{\epsilon}^{*}\alpha=\epsilon^{w}\alpha\}.
\end{eqnarray*}
Therefore Lie algebra cohomology splits $H^{q}(\mathfrak{g})=\bigoplus_{w}H^{q,w}(\mathfrak{g})$.

\begin{ex}
If $G=Heis^{2m+1}$ is the Heisenberg group, for each degree $q\not=0$, $2m+1$,
\begin{eqnarray*}
\Lambda^q \mathcal{G}^* =\Lambda^{q,q}\oplus\Lambda^{q,q+1},
\end{eqnarray*}
where $\Lambda^{q,q}=\Lambda^{q}(V^1 )^*$ and $\Lambda^{q,q+1}=\Lambda^{q-1}(V^1 )^* \otimes (V^2 )^*$.
\end{ex}

\textbf{Notation}:
$\displaystyle \Lambda^{q,\geq w}=\bigoplus_{w'\geq w}\Lambda^{q,w'}$. The space of differential forms which are smooth linear combinations of  left-invariant forms from $\Lambda^{q,\geq w}$ is denoted by $\Omega^{q,\geq w}$.

Note that each $\Omega^{\cdot,\geq w}$ is a differential ideal in the algebra of all differential forms $\Omega^\cdot$. 

\smallskip

%\textbf{Remark}: $\Omega^{*,\geq w}$ makes sense more generally for \emph{equiregular} sub-Riemannian manifolds (i.e. the distributions generated by successive brackets have constant rank).

\end{frame}

\begin{frame}

\begin{prop}
\label{uppernorm}
$G$ Carnot group. Let $U\subset G$ be a bounded open set with smooth boundary. Let $\omega$ be a closed differential form on $U$ of weight $\geq w$. Then, for every $\epsilon$ small enough, the cohomology class $\kappa\in H^q (U,\R)$ of $\omega$ can be represented by a straight cocycle $c_{\epsilon}$ (maybe defined on a slightly smaller homotopy equivalent open set) such that $|c_{\epsilon}|_{\epsilon}\leq \textrm{const.}\,\epsilon^w $. In other words, $\n{\kappa}_{w}<+\infty$.
\end{prop}

\proof
Use exponential map to push affine simplices in the Lie algebra to the group. Fill in all straight simplices in $G$ of unit size with such affine singular simplices $\sigma_1$. Apply $\delta_{\epsilon}$ and obtain a filling $\sigma_{\epsilon}$ for each straight simplex $\sigma$ in $G$ of size $\epsilon$. Define a straight cochain $c_{\epsilon}$ of size $\epsilon$ on $U$ by
\begin{eqnarray*}
c_{\epsilon}(\sigma)=\int_{\sigma_{\epsilon}}\omega.
\end{eqnarray*}
Since $\omega$ is closed, Stokes theorem shows that $c_{\epsilon}$ is a cocycle. Its cohomology class in $H^q (U',\R)\simeq H^q (U,\R)$ is the same as $\omega$'s. By compactess, if $\lambda\in\Lambda^{\cdot,w}$, $\displaystyle\sup_{\sigma_1}\int_{\sigma_1}\lambda\leq \mathrm{const.}(\lambda)$, so $\displaystyle\sup_{\sigma_\epsilon}\int_{\sigma_\epsilon}\lambda\leq \mathrm{const.}(\lambda)\,\epsilon^{w}$. Summing over components of $\omega$,
\begin{eqnarray*}
|c_{\epsilon}(\sigma)|
\leq\textrm{const.}(\omega)\,\epsilon^{w}.\qed
\end{eqnarray*}

\end{frame}

\subsection{Algebraic weights}

\begin{frame}

\begin{defi}
$G$ Carnot group. The \emph{algebraic weight} $AW_q (G)$ as the largest $w$ such that there exists arbitrarily small open sets with smooth boundary $U\subset G$ and nonzero classes in $H^q (U,\R)$ which can be represented by closed differential forms of weight $\geq w$.
\end{defi}

We just proved that $MW_q \geq AW_q$. 

\begin{cor}
Let $G$ be a Carnot group. Then for all $q=1,\ldots,n-1$, $\alpha(G)\leq\frac{q}{AW_q}$.
\end{cor}

So our goal now is to show that for certain Carnot groups, for certain degrees $q$, in every open set, every closed differential $q$-form is cohomologous to a form of high weight.

\end{frame}

\section{Rumin's complex}

\begin{frame}

\subsection{Algebraic weight estimate}

It turns out that the obstruction for cohomologing $q$-forms towards weight $>w$ is $H^{q,w}(\mathfrak{g})$.

\begin{thm}[\red{Rumin 2005}]
Let $G$ be a Carnot group. Assume that, in the cohomology of the Lie algebra, $H^{q,w'}(\mathfrak{g})=0$ for all $w'<w$. Then $AW_q (G)\geq w$.
\end{thm}\pause

This will be proven later. First reformulate and illustrate. On Carnot groups, the grading of cohomology is compatible with Poincar\'e duality, $H^{q,w}(\mathfrak{g})=H^{n-q,Q-w}(\mathfrak{g})$. So 
\begin{eqnarray*}
H^{n-q}(\mathfrak{g})=H^{n-q,\leq Q-w}(\mathfrak{g})\quad\Rightarrow\quad AW_q (G)\geq w.
\end{eqnarray*}\pause

\begin{ex}
Degree $n-1$. On any Carnot Lie algebra $\mathfrak{g}$, closed $1$-forms belong to $(V^1)^* =\Lambda^{1,1}$, so $H^1 (\mathfrak{g})=H^{1,1}(\mathfrak{g})$, and $AW_{n-1}(M)\geq Q-1$.
\end{ex}


From now on, the source is Rumin's lecture notes \emph{An introduction to spectral and differential geometry in Carnot-Carath\'eodory spaces}.

\end{frame}

\begin{frame}

\begin{ex}
$2m+1$-dimensional contact manifolds. Closed $m$-forms belong to $\Lambda^{m,m}$. Therefore $H^m (\mathfrak{g})=H^{m,m}(\mathfrak{g})$, $AW_{m+1}(M)\geq m+2$ and $\alpha\leq\frac{m+1}{m+2}$.
\end{ex}
Indeed, if $\omega\in\Lambda^{m,m+1}$, $\omega=\theta\wedge\phi$ where $\theta\in (V^2)^*$, $\phi\in\Lambda^{m-1,m-1}$, $(d\omega)^{m+1,m+1}=(d\theta)\wedge\phi\not=0$ since $d\theta$ is symplectic on $\Delta$.
\pause

%\begin{ex}
%Generic sub-Riemannian case. Let $h=dim(\Delta)$ and $k$ be such that $h-k\geq(n-h)k$. Then $H^{k}(\mathfrak{g}_m)=H^{k,k}(\mathfrak{g}_m)$, thus $AW(M)\geq Q-k$.
%\end{ex}
%Let $\theta$ be a $\R^{n-h}$-valued $1$-form defining $\Delta$. Say a $k$-plane $S\subset \Delta_m$ is {\em isotropic} if $d\theta_{|S}=0$. Say $S$ is {\em regular} if the map $\Delta_m \to Hom(S,\R^{n-h})$, $u\mapsto (\iota_u d\theta)_{|S}$ is onto. $h-k\geq(n-h)k$ is a necessary condition for existence of regular isotropic horizontal $k$-planes. It is generically sufficient. When it holds, closed left-invariant $k$-forms have to be of weight $k$, so $H^{k}(\mathfrak{g}_m)=H^{k,k}(\mathfrak{g}_m)$ for $m\in M$.

\begin{ex}
Engel case. The nonzero weight spaces in $H^{\cdot}(\mathfrak{g})$ are $H^{0,0}$, $H^{1,1}$, $H^{2,3}$, $H^{2,4}$, $H^{3,6}$ and $H^{4,7}$. So the best bound on $\alpha(G)$ is $\frac{1}{2}$, achieved for degree $3$. Disappointing.
\end{ex}\pause

\begin{ex}
Quaternionic Heisenberg group $Heis_{\H}^{4m-1}$. 
\red{Julg 1995} shows that $H^{q}(\mathfrak{g})=H^{q,\geq q+2}$ if $q\geq 2m$ and $H^{q}(\mathfrak{g})=H^{q,\geq q+3}$ if $q\geq 3m$.
% $H^{1,1}$, $H^{2,2}$, $H^{2,3}$, $H^{3,4}$, $H^{4,6}$, $H^{5,7}$, $H^{5,8}$,$H^{6,9}$ and $H^{7,10}$. 
Thus $\alpha(G)\leq\frac{2m}{2m+2}=\frac{3m}{3m+3}$, obtained when considering degrees $2m$ and $3m$.
%$\frac{2}{3}$, achieved for degrees $4$ and $6$. 
\end{ex}\pause

The method just exposed seems to cover all presently known results on the H\"older homeomorphism problem.

\end{frame}

\subsection{Rumin's construction}

\begin{frame}

\proof \textbf{of Rumin's theorem in degree $n-1$}: reformulation of argument in proof of isoperimetric inequality. 
\smallskip

Notation: $d_0=$ exterior differential on left-invariant forms, seen as a 0-order differential operator on differential forms. \pause

Write $\omega=\omega_{<Q-1}+\omega_{Q-1}$. $d_0=0$ in degree $n-1$ and $H^{n-1}(\mathfrak{g})=H^{n-1,Q-1}$ imply that $\omega_{<Q-1}\in\mathrm{im}(d_0)$. Pick a linear inverse $d_{0}^{-1}$ on $\mathrm{im}(d_0)$. Consider $r\omega=\omega-dd_{0}^{-1}(\omega_{<Q-1})$. Then weight$(r\omega)>$ weight$(\omega)$, unless $\omega_{<Q-1}=0$. Therefore iterating leads to a cohomologous form of weight $Q-1$.\pause

\bigskip

\textbf{General case}: not all left-invariant $q$-forms are closed. Instead of considering $1-dd_{0}^{-1}$ on closed forms, construct a homotopy of chain complexes $r=1-dd_{0}^{-1}-d_{0}^{-1}d$.\pause

Pick homogeneous complements $W_0$ of $\mathrm{ker}(d_0)$ in $\Lambda^{\cdot}\mathfrak{g}^*$, and $E_0$ of $\mathrm{im}(d_0)$ in $\mathrm{ker}(d_0)$. 
\begin{eqnarray*}
\Lambda^{\cdot}\mathfrak{g}^*=\mathrm{im}(d_0)\oplus E_0\oplus W_0.
\end{eqnarray*}
Set $d_{0}^{-1}=0$ on $E_0\oplus W_0$ and extend to $\Lambda^{\cdot}\mathfrak{g}^*$ using the inverse of $d_0:\mathrm{im}(d_0)\to W$.  

Denote by $\pi_0:\Lambda^{\cdot}\mathfrak{g}^*\to E_0$ the projector. $d_{0}^{-1}$ and $\pi_0$ extend to differential forms, denote by $\mathcal{E}_0=\mathrm{im}(\pi_0)$.
\end{frame}

\begin{frame}

\begin{thm}[\red{Rumin 1999}]
Let $r=1-dd_{0}^{-1}-d_{0}^{-1}d$. Iterates of $r$ stabilise to a differential operator $\pi$ which is again a homotopy equivalence of chain complexes. $\pi$ is a projector onto the subcomplex
\begin{eqnarray*}
\mathcal{E}=\mathrm{ker}(d_0^{-1})\cap \mathrm{ker}(d_0^{-1}d).
\end{eqnarray*}
The restriction of $\pi_0$ to $\mathcal{E}$ is an isomorphism onto $\mathcal{E}_0$, with inverse the restriction of $\pi$ to $\mathcal{E}_0$.
\end{thm}\pause

\textbf{Corollary}: estimate of algebraic weight follows. If $\omega$ is closed, $\pi(\omega)=\pi\circ\pi_0\circ \pi(\omega)$ is cohomologous to it. The weights present is $E_0=\mathrm{im}(\pi_0)$ are those of the cohomology. $\pi$, like $d$, does not decrease weights. Thus weight$(\pi(\omega))$ is as high as the minimum weight in cohomology.\pause

\medskip

\textbf{Example}: $Heis$.

$\mathrm{ker}(d_0)=\mathrm{span}\{1,dx,dy,dx\wedge dy,dx\wedge\theta,dy\wedge\theta,dx\wedge dy\wedge\theta\}$.

$\mathrm{im}(d_0)=\mathrm{span}\{dx\wedge dy\}$.

$W_0=\mathrm{span}\{\theta\}$.

$E_0=\mathrm{span}\{1,dx,dy,dx\wedge\theta,dy\wedge\theta,dx\wedge dy\wedge\theta\}$.

No need to iterate. $\pi=r$ maps $adx+bdy+c\theta$ to $adx+bdy+(Ya-Xb)\theta$ and $edx\wedge dy+fdx\wedge\theta+gdy\wedge\theta$ to $(Xe+f)dx\wedge\theta+(Ye+g)dy\wedge\theta$.

\end{frame}

\subsection{Proof of Rumin's assertions}

\begin{frame}

\proof  1. \emph{Stabilization}. By construction, $r=1$ on $\mathcal{E}$. Let $\mathcal{W}$ denote the space of differential forms which belong to $W$ at each point. Then, on $W$, $r$ is nilpotent. Indeed, $d_{0}^{-1}=0$ on $W$ so $r=1-d_{0}^{-1}d=-d_{0}^{-1}(d-d_0)$ maps $\mathcal{W}$ to itself and strictly increases weight. Since $rd=dr$, the same is true on $d\mathcal{W}$ and thus on $\mathcal{F}:=\mathcal{W}+d\mathcal{W}$.\pause

\smallskip

2. \emph{Claim}: $\Omega^{\cdot}=\mathcal{E}\oplus\mathcal{F}$.

Since $N=d_{0}^{-1}(d-d_0)_{|\mathcal{W}}$ is nilpotent, $1+N$ has a differential inverse $P=1+\sum(-1)^i N^i$ defined on $\mathcal{W}$. Set $Q=Pd_0^{-1}:\Omega^{\cdot}\to\mathcal{W}$. We check that $\pi=1-Qd-dQ$ is the projector onto $\mathcal{E}$ with kernel $\mathcal{F}$. By definition, $d_0^{-1}(\mathcal{E})=0$, $d_0^{-1}d(\mathcal{E})=0$ so $\pi=1$ on $\mathcal{E}$. Also $d_0^{-1}Q=0$, $d_0^{-1}dQ=(1+N)Pd_0^{-1}=d_0^{-1}$ so $d_0^{-1}\pi=0$. Since $d\pi=\pi d$, $d_0^{-1}d\pi=0$ so $\mathrm{im}(\pi)\subset\mathcal{E}$, thus $\pi$ is a projector onto $\mathcal{E}$. Use $\mathrm{ker}(\pi)=\mathrm{im}(dQ+Qd)$. $\mathrm{im}(Q)\subset\mathcal{W}$ so $\mathrm{im}(dQ)\subset\mathcal{dW}$, and $\mathrm{im}(Qd+dQ)\subset\mathcal{W}+d\mathcal{W}$. Conversely, on $\mathcal{W}$, $Qd=Pd_0^{-1}d=1$ so $\mathcal{W}\subset\mathrm{ker}(\pi)$. Since $d\pi=\pi d$, $d\mathcal{W}\subset\mathrm{ker}(\pi)$, so $\mathrm{ker}(\pi)=\mathcal{F}$.\pause

\smallskip

3. \emph{$\pi$ and $\pi_0$ are inverses of each other on $\mathcal{E}_0$ (resp. $\mathcal{E}$)}. Since $\mathrm{im}(\pi_0)=\mathcal{E}_0\subset\mathrm{ker}(d_0^{-1})\subset\mathrm{ker}(Q)$, $Q\pi_0=0$. Since $\mathrm{im}(Q)\subset \mathcal{W}\subset\mathrm{ker}(\pi_0)$, $\pi_0 Q=0$. Thus $\pi_0\circ\pi\circ\pi_0=\pi_0(1-Qd-dQ)\pi_0=\pi_0$, i.e. $\pi_0\circ\pi_{|\mathcal{E}_0}=1$. Since $\mathcal{E}\subset\mathrm{ker}(d_0^{-1})=\mathcal{W}\oplus\mathcal{E}_0$ and $\mathcal{W}\subset\mathrm{ker}(\pi_0)$, $\mathrm{ker}({\pi_0}_{|\mathcal{E}})=\mathcal{W}\cap\mathcal{E}=\{0\}$, so ${\pi_0}_{|\mathcal{E}}$ is injective, and ${\pi\circ\pi_0}_{|\mathcal{E}}=1$.

\end{frame}

\subsection{Further applications of Rumin's complex}

\begin{frame}

\begin{defi}
The differential operator $d_c=\pi_0 d\pi$ on $\mathcal{E}_0$ is \textrm{Rumin's complex}. Computes cohomology.
\end{defi}

\textbf{Example}: $Heis$. $\mathcal{E}_0^0=\Omega^0$, $\mathcal{E}_0^3=\Omega^30$, $\mathcal{E}_0^1=\mathrm{span}\{dx,dy\}$, $\mathcal{E}_0^2=\mathrm{span}\{dx\wedge\theta,dy\wedge\theta\}$. On forms of degrees 0, 2 and 3, $d_c=d$.
On 1-forms, $d_c$ is a second order operator, $d_c(adx+bdy)=(YXa-XXb)dx\wedge\theta+(-XYb+YYa)dy\wedge\theta$.\pause 

\medskip

\textbf{Applications}

\red{Rumin 1994, 2000, Ge 1994}. Asymptotics for the spectrum of the Laplace-Beltrami operator on differential forms for sequences of Riemannian metrics converging to a sub-Riemannian metric.

\red{Julg-Kasparov 1995}. Baum-Connes conjecture for $SU(n,1)$.

\red{Rumin 2002}. Estimates on Novikov-Shubin invariants of nilpotent groups (large scale spectral invariants).

\red{Julg 2002}. Baum-Connes conjecture for $Sp(n,1)$.

\red{Biquard-Herzlich-Rumin 2006}. Expressing secondary invariants of compact $CR$-manifolds.

\end{frame}

\section{Quasisymmetric Hölder-Lipschitz equivalence problem}

\begin{frame}

\begin{center}
Lecture 4
\end{center}
\begin{center}
\textbf{Quasisymmetric H\"older equivalence problem}
\end{center}

\end{frame}

\begin{frame}

Recall Gromov's slogan:

\hbox{\vbox{\hsize=5cm \begin{center}
\begin{tabular}{|c|}
\hline
Here\\
\hline
Euclidean\\
\hline
$\exists$ $k$-dimensional subset with\\
Hausdorff dimension $\leq k$\\
\hline
\end{tabular}
\end{center}}\hfill\vbox{\hsize=5cm \begin{center}
\begin{tabular}{|c|}
\hline
There \\
\hline
Carnot\\
\hline
$\forall$ $k$-dimensional subset,\\
Hausdorff dimension $\geq k'$\\
\hline
\end{tabular}
\end{center}}}
\centerline{then $\alpha(Carnot)\leq\frac{k}{k'}.$}\pause

\medskip

Next is a result where

\hbox{\vbox{\hsize=5cm \begin{center}
\begin{tabular}{|c|}
\hline
Here\\
\hline
Euclidean\\
\hline
explicit family of subsets with\\
Hausdorff dimension $\leq k$\\
\hline
\end{tabular}
\end{center}}\hfill\vbox{\hsize=5cm \begin{center}
\begin{tabular}{|c|}
\hline
There \\
\hline
Carnot\\
\hline
for almost every image subset,\\
Hausdorff dimension $\geq k'$\\
\hline
\end{tabular}
\end{center}}}
\centerline{then $\alpha(Carnot)\leq\frac{k}{k'}$.}\pause

\bigskip

Furthermore, the argument uses quasisymmetric invariants, whence
\centerline{$\alpha_{qs}(Carnot)\leq\frac{k}{k'}$.}

\end{frame}

\subsection{Coarea ?}

\begin{frame}

Let $f:\R^n\to G$ be a H\"older homeomorphism with Lipschitz $f^{-1}:G\to\R^n$. Let $v:\R^n\to\R$ be a coordinate function. Then $u=v\circ f^{-1}$ is Lipschitz. Imagine that \emph{coarea inequality} holds:

\begin{eqnarray}
\label{coareaineq}
\int_{G}Lip_u^Q\leq\int_{\R}\left(\int_{u^{-1}(t)}Lip_u^{Q-1}\right)\,dt\leq \mathrm{const.}\,\int_{\R}\mathcal{H}^{Q-1}(u^{-1}(t))\,dt.
\end{eqnarray}
Here, $Lip_u$ denotes the local Lipschitz constant. Since, for non constant $u$, $\int_{X}Lip_u^Q>0$, this shows that there exists $t\in\R$ such that $\mathcal{H}^{Q-1}(u^{-1}(t))>0$, and therefore $u^{-1}(t)$ has Hausdorff dimension at least $Q-1$. \pause

\bigskip

Unfortunately, \red{Magnani 2002}'s \emph{coarea inequality} goes in the opposite direction!\pause

\bigskip

\textbf{Strategy}: replace conformally invariant integrals $\int Lip_u^Q$ with \emph{packing measures} which are quasisymmetric invariants and satisfy coarea inequality in the right direction. If possible, use vector valued function $u$.

\end{frame}

\subsection{Packing measures}

\begin{frame}

Let $N$ be an integer, let $\ell\geq 1$. Let $X$ be a metric space. An \emph{$(N,\ell)$-packing} is a countable collection of balls $\{B_j\}$ such that the collection of concentric balls $\ell B_j$ has multiplicity $< N$.\pause

\smallskip

Let $\phi$ be a positive function on the set of balls in $X$. Define
$$\Phi_{N,\ell}^{p;\epsilon}(A)=\sup\{\sum_{i}\phi(B_{i})^p\,;\,\{B_{i}\}\,(N,\ell)-\textrm{packing of }X,\textrm{ centered on }A,\textrm{ of mesh }\leq\epsilon\}.$$

Define the \emph{packing pre-measure associated to $\phi$} by
$$\Phi_{N,\ell}^{p}(A)=\lim_{\epsilon\to 0}\Phi_{N,\ell}^{p;\epsilon}(A).$$\pause

\bigskip

\textbf{Example}: $\pi(B)=$radius$(B)$ leads to ``usual'' packing measure $\Pi^p$ and packing dimension. It is H\"older covariant.

\bigskip

\textbf{Example}: let $u:X\to M$ measure space. $e_u(B)=$measure$(B)$ leads to \emph{$p$-energy} $E_u^{p}$. It is quasisymmetry invariant: if $g$ is quasisymmetric, $\forall\ell$, $\exists\ell'$ such that
$$E_{u,N,\ell}^{p}(A)\leq E_{u',N,\ell'}^{p}(g(A)).$$

\end{frame}

\subsection{Coarea inequality}

\begin{frame}

\begin{prop}
\label{coarea}
Let $X$ be a metric space. Let $u:X\to M$ be a map to a measure space $(M,\mu)$. Then
\begin{eqnarray*}
E_{u}^{p}(X)\leq \int_{M}E_{u}^{p-1}(u^{-1}(m))\,d\mu(m).
\end{eqnarray*}
\end{prop}
\begin{eqnarray*}
\sum_{i}\mu(u(B_i))^p 
&=&\sum_{i}(\int_{M}1_{u(B_{i})}(m)\,d\mu(m))\mu(u(B_i))^{p-1}\\
&=&\int_{M}(\sum_{i}1_{u(B_{i})}(m)\mu(u(B_i))^{p-1})\,d\mu(m)\\
&=&\int_{M}(\sum_{\{i\,;\,m\in u(B_{i})\}}\mu(u(B_i))^{p-1})\,d\mu(m)\\
&\leq&\int_{M}E_u^{p-1;2\epsilon}\,d\mu(m).\qed
\end{eqnarray*}

If $u$ is Lipschitz and $\mu(B)\sim$radius$(B)^d$ in $M$, $E_u^{p-1}\leq\Pi^{d(p-1)}$, thus
\begin{eqnarray*}
E_{u}^{p}>0\quad\Rightarrow\quad\exists m,\,\mathrm{dim}(u^{-1}(m))\geq d(p-1).
\end{eqnarray*}
\end{frame}

\subsection{Positivity of energy}

\begin{frame}

\begin{corollary}
Let $f:\R^n\to G$ be a $C^{\alpha}$-H\"older homeomorphism, with Lipschitz$f^{-1}$. If $v:\R^n\to\R^d$ is a submersion and $u=v\circ f^{-1}$ satisfies $E_u^{Q/d}>0$, then $\displaystyle\alpha_{qs}(G)\leq\frac{n-d}{Q-d}$.
\end{corollary}\pause

\begin{prop}
If $u:G\to\R$ is continuous and non constant, $E_u^Q>0$.
\end{prop}
\textbf{Corollary}: $\alpha_{qs}(G)\leq \frac{n-1}{Q-1}$.

\proof Length-area method. Let $\Gamma$ denote the family of unit segments parallel to a side of unit $n$-cube. For all positive functions $\rho$ on the square,
\begin{eqnarray*}
\int_{0}^{1}(\int_{\gamma}\rho\,ds)\,d\gamma\leq\int\rho^n\,dx_1\ldots dx_n.
\end{eqnarray*}
Replace integrals with packing (resp. covering) measures, apply to family of parallel horizontal line segments in $G$.

\end{frame}

\begin{frame}

\begin{prop}
Let $X$ be a metric space. Let $\Gamma$ be a family of subsets of $X$, equipped with a measure $d\gamma$. For each $\gamma\in\Gamma$, a probability measure $m_{\gamma}$ is given on $\gamma$. Let $p\geq 1$. Assume that 
\begin{eqnarray*}
\int_{\{\gamma\in\Gamma\,;\,\gamma\cap B\not=\emptyset\}}m_{\gamma}(\gamma\cap \ell B)^{1-p}\,d\gamma\leq \tau.
\end{eqnarray*}
Then, for every function $\phi$ on the set of balls of $X$,
\begin{eqnarray*}
\Phi^{p}(X)\geq\frac{1}{\tau}\int_{\Gamma}\tilde{\Phi}^{1}(\gamma)^p\,d\gamma.
\end{eqnarray*}
\end{prop}

\proof Let $1_i(\gamma)=1$ iff $\gamma\cap B_i\not=\emptyset$. The balls such that $1_i(\gamma)=1$ cover $\gamma$, thus
\begin{eqnarray*}
\tilde{\Phi}^{1;\epsilon}(\gamma)\leq\sum_{i}\phi(B_i)1_{i}(\gamma)=\sum_{i}\phi(B_i)1_{i}(\gamma)m_\gamma(\gamma\cap\ell B_i)^{\frac{1-p}{p}}m_\gamma(\gamma\cap\ell B_i)^{\frac{p-1}{p}}.
\end{eqnarray*}
H\"older's inequality gives
\begin{eqnarray*}
\tilde{\Phi}^{1;\epsilon}(\gamma)^p&\leq&\left(\sum_{i}\phi(B_i)^p 1_{i}(\gamma)m_\gamma(\gamma\cap\ell B_i)^{1-p}\right)\left(\sum_{i}m_\gamma(\gamma\cap\ell B_i)\right)^{p-1}.
\end{eqnarray*}
Integrate over $\Gamma$. 

\end{frame}

\begin{frame}

\begin{center}
\begin{tabular}{ccc}
  closed $n-1$-form &$\Leftrightarrow$& map to $\R^{n-1}$\\
  &&\\
    closed $n-1$-form of weight $Q-1$&$\Leftrightarrow$&map to $\R^{n-1}$ with horizontal fibers\\
\end{tabular}
\end{center}
However, map to $\R^{n-1}$ is not Lipschitz in general.
\bigskip

On $Heis^3$, weight 3 2-forms give rise to cocycles such that $E^{4/3}<\infty$ thus $E^{2}=0$. But these correspond to maps which are not Lipschitz at all.

\begin{ques}
Let $X$ be a metric space which is quasisymmetric to an open subset of $Heis^3$. Let $u:X\to\R^2$ be Lipschitz and open. Show that $E^{p}_{u}>0$ for all $p<2$.
\end{ques}
This would follow from
\begin{ques}
Let $u:Heis\to\R^2$ be continuous and open. Assume that both components $u^1$ and $u^2$ satisfy $\tilde{E}_{u_i}^4<\infty$. Show that $u$ is a.e. differentiable and infer that $E^{2}_{u}>0$.
\end{ques}

\end{frame}

\end{document}