\documentclass[8pt]{beamer}
\usepackage[ansinew]{inputenc}
\usepackage{amssymb, color}
\usepackage{graphicx}
\usepackage{beamerthemesplit}

\title{The quasisymmetric Hölder equivalence problem}
\author{P. Pansu}
\date{September 2nd, 2021}

\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}

\section{Where my interest in subRiemannian geometry stems from}

\subsection{Complex hyperbolic plane}

\begin{frame}

The unit ball $B$ in $\C^2$ is the prototype of
\begin{itemize}
  \item a symplectic manifold with convex boundary;
  \item a complex manifold with strictly pseudoconvex boundary.
\end{itemize}
Furthermore, it carries a Riemannian metric which is invariant under all holomorphic automorphisms. It becomes \emph{complex hyperbolic plane} $H^2_\C$.\pause ~The boundary $\partial B$ carries subRiemannian metrics, permuted by automorphisms, which are conformal to one another.\pause

\medskip

\begin{minipage}{5.5cm}
In $B$, there are two types of totally geodesic surfaces:
\begin{itemize}
  \item complex lines, with curvature $-1$,
  \item lagrangian planes, with curvature $-\frac{1}{4}$.
\end{itemize}
Lagrangian planes intersect the boundary along Legendrian curves. Therefore the contact structure on the boundary is determined by the Riemannian geometry inside the ball. \end{minipage}\hskip1cm\begin{minipage}{4cm}\includegraphics[width=1.5in]{spinal.png}

\hfill \tiny{W. Goldman}\end{minipage}\pause

\medskip

What if one perturbs the Riemannian metric boundedly (i.e. in a biLipschitz manner), while keeping curvature negative? Is there a smooth boundary? With a contact structure?

\end{frame}

\subsection{Boundaries of negatively curved manifolds}

\begin{frame}

\textbf{Facts}. \begin{itemize}
  \item Negatively curved manifolds $M$ have a visual sphere $\partial M$, equipped with visual metrics (one for each point of $M$). Two such metrics are conformally related.
  \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}\pause

So, the visual sphere $\partial M$ is a metric space, it is not smooth, it merely has a quasisymmetric (weaker than biLipschitz) structure. And a H\"older structure too. One cannot make sense of a contact structure on it.

\end{frame}

\section{The quasisymmetric H\"older 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 equivalent to a $\delta$-pinched complete simply connected Riemannian manifold. \pause 
\end{defi}

Example: Real hyperbolic spaces have $\delta(H^n_\R)=-1$. Complex hyperbolic plane $H^{2}_{\C}$ is $-\frac{1}{4}$-pinched, so $\delta(H^{2}_{\C})\leq -\frac{1}{4}$. Is it true that $\delta(H^{2}_{\C})=-\frac{1}{4}$ ?\pause   ~Open.\pause

\bigskip

\textbf{Approach using boundary homeomorphisms:}
\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}

\medskip

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

\medskip

\textbf{Claim}. $\sqrt{-\delta(M)}\leq \alpha_{qs}(\partial M)$. Proof on next slide.

\end{frame}

\begin{frame}

\textbf{Recall definitions}

\medskip

\begin{minipage}{4cm}
$\delta(M)=\inf$ pinching of Riemannian manifolds $N$ biLipschitz to $M$
\end{minipage}\hskip1cm \begin{minipage}{4cm}
$\alpha_{qs}(\partial M)=\sup$ H\"older exponent $\alpha$ of homeos of round sphere $S\to X$ with Lipschitz inverse, with $X$ quasisymmetric to $\partial M$. 
\end{minipage}

\medskip

\textbf{Proof}. 

$$
N \text{ biLipschitz to }N \implies \partial M \text{ quasisymmetric to }\partial N.
$$

\begin{align*}
N ~\delta-\text{pinched} \implies &\exists C^{\alpha} \text{ homeomorphism with Lipschitz inverse from}\\
& \text{round sphere to } \partial N  \text{ with } \alpha=\sqrt{-\delta}.
\end{align*}

\textbf{Conclusion}. $\sqrt{-\delta(M)}\leq \alpha_{qs}(\partial M)$.

\end{frame}

\begin{frame}

\begin{thm}[\textcolor{red}{Pansu 2020}]
For all sub-Riemannian $n$-manifolds $X$ with Hausdorff dimension $Q$, $\alpha_{qs}(X)\leq\frac{n-1}{Q-1}$. 

In particular, $\alpha_{qs}(Heis)\leq\frac{2}{3}$.
\end{thm}\pause

\medskip

(The consequence $\delta(H^2_{\C})\geq -\frac{4}{9}$ was already known).

\end{frame}

\subsection{H\"older equivalence problem}

\begin{frame}
Remark: The obvious map $\R^3\to Heis=$ subRiemannian Heisenberg group is $C^{1/2}$-H\"older continuous.

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

\begin{defi}
Let $\alpha(X)=\sup\{\alpha\in(0,1)\,|\,\exists$ locally a $C^\alpha$ homeomorphism $\R^n \to X\}$.
\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. $X$ have dimension $n$, Hausdorff dimension $Q$. Then $\alpha(X)\leq\frac{n-1}{Q-1}$.

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

\end{frame}

\subsection{Weakly differentiable contact mappings}

\begin{frame}

Given a contact structure $\xi$, pick a smooth sub-Riemannian metric $d$. Then locally, $d$ is unique up to biLipschitz equivalence. Local becomes global on compact manifolds. Every biLipschitz invariant notion becomes a concept in contact topology.\pause

\bigskip

The following classes of homeomorphisms can be thought of as classes of weakly differentiable contactomorphisms:
\begin{enumerate}
  \item BiLipschitz maps.
  \item \begin{minipage}{4cm}Quasisymmetric maps.\end{minipage}\begin{minipage}{4cm}\includegraphics[width=2in]{qs.pdf}\end{minipage}\pause
  \item Sobolev $L_1^p$ maps for $p$ close to $4$ (how close?).
  \item H\"older $C^\alpha$ maps for $\alpha<1$ close to $1$ (how close?).
\end{enumerate}\pause

\begin{ques}
What is the classification of contact structures in each of these categories?
\end{ques}

\end{frame}

\subsection{$1/2$ or $2/3$?}

\begin{frame}

Remark: The obvious map $\R^3\to Heis=$ subRiemannian Heisenberg group is $C^{1/2}$-H\"older continuous, and its inverse is Lipschitz, so $\alpha(Heis)\geq\frac{1}{2}$.

Therefore, if $\alpha\leq \frac{1}{2}$, $C^{\alpha}$ maps are meaningless for contact geometry.

\begin{thm}[\red{Gromov 1993}]
$\alpha(Heis)\leq\frac{2}{3}$.
\end{thm}

\begin{thm}[\red{Wenger-Young 2018}]
There exist proper degree $1$ $C^\alpha$ maps $\R^3\to Heis$ for all $\alpha<\frac{2}{3}$.
\end{thm}

So perhaps $C^{\alpha}$ maps are meaningless for contact geometry as soon as $\alpha\leq \frac{2}{3}$. Who knows?\pause

\medskip

\begin{ques}
If two compact contact $3$-dimensional contact manifolds are $C^\alpha$ biH\"older-equivalent, for $\alpha\ge\frac{2}{3}$, are they contactomorphic?
\end{ques}

\end{frame}

\begin{frame}

\textbf{Remark}. The cohomology of a subRiemannian manifold is filtered by weights
\begin{itemize}
  \item either by weights of differential forms,
  \item or by weights of Alexander-Spanier cochains.
\end{itemize}

Both are easily biLipschitz invariant.

The first one is finer.

The second filtration is $C^\alpha$ biH\"older-invariant, for $\alpha$ close enough to $1$.

\begin{ques}
Show that the first one is invariant under Sobolev $L_1^p$ homeomorphisms, for $p$ close enough to $Q$.
\end{ques}
This convers quasisymmetric homeomorphisms.

Unfortunately, the filtration is trivial for contact manifolds (Rumin). 

\medskip

Thus the contact case seems the hardest.

\end{frame}

\begin{frame}

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
SubRiemannian\\
\hline
$\forall$ $k$-dimensional subset,\\
Hausdorff dimension $\geq k'$\\
\hline
\end{tabular}
\end{center}}}
\centerline{then $\alpha(SubRiemannian)\leq\frac{k}{k'}.$}\pause

\medskip

In my argument,

\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
SubRiemannian\\
\hline
for almost every image subset,\\
Hausdorff dimension $\geq k'$\\
\hline
\end{tabular}
\end{center}}}
\centerline{then $\alpha(SubRiemannian)\leq\frac{k}{k'}$.}\pause

\bigskip

Furthermore, only quasisymmetric invariants are used, whence
\centerline{$\alpha_{qs}(SubRiemannian)\leq\frac{k}{k'}$.}

\end{frame}

\section{Coarea inequality}

\begin{frame}

The \emph{coarea formula} amounts to integrating along the level sets of a Lipschitz function $u:\R^n\to\R$. For every positive function $\phi$ on $\R^n$,

\vskip-.5cm

\begin{minipage}{5cm}
$$\int_{\R^n}\phi=\int_{\R}(\int_{u^{-1}(t)}\frac{\phi}{|\nabla u|})\,dt$$
\end{minipage} \begin{minipage}{5cm}\includegraphics[height=2in]{coarea.pdf}\end{minipage}\pause

\vskip-1cm

A special case: $\phi=|\nabla u|^n$.
$$
\int_{X}|\nabla u|^n=\int_{\R}(\int_{u^{-1}(t)}|\nabla u|^{n-1})\,dt.
$$
Both sides are conformally invariant.

\end{frame}

\begin{frame}

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

\begin{eqnarray}
\label{coareaineq}
\int_{X}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 Whereas $f^{-1}(u^{-1}(t))=v^{-1}(t)$ has Hausdorff dimension $n-1$. This implies that $\alpha\leq\frac{n-1}{Q-1}$.

\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 energy measures} which are quasisymmetric invariants and satisfy coarea inequality in the right direction. If possible, extend to vector valued maps $u$.

\end{frame}

\section{A large scale question}

\begin{frame}

Here is a global question about noncompact contact manifolds.

\medskip

A map $f:X\to Y$ between metric spaces is a \emph{quasiisometry} if it is onto (up to bounded distance) and satisfies the biLipschitz inequality
$$
d(f(x),f(x'))\le L d(x,x')
$$
for distances $d(x,x')\ge 1$.\pause

\medskip

As P. Massot explained, on a contact manifold, every subRiemannian metric determines a Riemannian metric. Say a subRiemannian metric has \emph{bounded geometry} if the corresponding Riemannian metric has bounded geometry (unit balls are uniformly biLipschitz to Euclidean unit balls).\pause

If so, both metrics are quasiisometric.\pause

\begin{ques}
Given a bounded geometry Riemannian $3$-manifold $M$ and a contact structure $\xi$ on $M$, does there exist a subRiemannian metric (subordinate to $\xi$) of bounded geometry which is quasiisometric to $M$ ?
\end{ques}

\end{frame}

\begin{frame}

\textbf{Approach in case $M$ is diffeomorphic to $\R^3$}

\medskip

Given a bounded geometry Riemannian metric, construct a contact structure with bounded covariant derivatives (a controlled Lutz theorem). 

Then the subRiemannian metric obtained by restriction has bounded geometry and is quasiisometric to given Riemannian metric. 

\medskip

This gives a positive answer for the contact structure which is overtwisted in all neighborhoods of infinity.

\medskip

How can one get the tight contact structure?

\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}

\end{document}