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\title{$L^p$-cohomology and curvature pinching}
\author{P. Pansu}
\date{\today}

\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{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 |}}

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

\begin{document}


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

\section{Gromov's H\"older homeomorphism problem}

\begin{frame}

\begin{defi}
A {\em Carnot group} is a Lie group $G$ admitting a one-parameter group $\delta_t$ of automorphisms, generated by a derivation whose bottom eigenspace generates the Lie algebra. $G$ carries left invariant {\em subRiemannian} (or {\em Carnot-Caratheodory}) metrics $d$ such that $d\circ\delta_t =td$.
\end{defi}%\pause

\begin{ques}
\textcolor{red}{\em (Gromov 1993)}. Let $G$ be an $n$-dimensional Carnot group. For which $\alpha\in(0,1)$ does there exist locally a homeomorphism $\R^n \to G$ which is $C^{\alpha}$-H\"older continuous ? I.e. compute $\alpha(G)=\sup\{\alpha\in(0,1)\,|\,\exists$ locally a homeomorphism $\R^n \to G\}$.
\end{ques}%\pause

\begin{ex}
If $G$ is a $r$-step Carnot group, the exponential map $\mathfrak{g}=Lie(G)\to G$ is locally $C^{1/r}$-H\"older continuous. Thus $\alpha(G)\geq 1/r$.
\end{ex}%\pause

\begin{prop}
Let $M$ have Hausdorff dimension $Q$. Then $\alpha(M)\leq\frac{n}{Q}$.
\end{prop}

\end{frame}

\subsection{Isoperimetric inequality}

\begin{frame}

\begin{prop}
Let $G$ be a Carnot group of dimension $n$ and Hausdorff dimension $Q$. Then $\alpha(M)\leq\frac{n-1}{Q-1}$.
\end{prop}%\pause

\def\dh{d^{H}}

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

\begin{frame}

\begin{prop}
\textcolor{red}{\em (Gromov 1993)}. Let $\mathbb{H}^{m}$ denote $2m+1$-dimensional Heisenberg group. Let $V\subset\mathbb{H}^{m}$ be a subset of topological dimension $m+1$. Then the Hausdorff dimension of $V$ is at least $m+2$. It follows that $\alpha(\mathbb{H}^{m})\leq\frac{m+1}{m+2}$.
\end{prop}%\pause

\textbf{Proof}. According to topological dimension theory (Alexandrov), there exists an $m$-dimensional polyhedron $P$ and a continuous map $f:P\to\mathbb{H}^{m}$ such that every map sufficiently $C^0$-close to $f$ hits $V$. 

\medskip

Gromov approximates $f$ with piecewise {\em horizontal} maps which sweep an open set $U$. This gives rise to a local projection $p:U\to\R^{m+1}$ such that for every ball $B$, the tube $p^{-1}(p(B))$ has volume $\leq$ const. $\mathrm{diameter}(B)^{m+2}$. 

\medskip

Cover $V$ with balls $B_j$. The corresponding tubes $T_j = p^{-1}(p(B_j))$ cover $U$. Then the volume of $U$ is less than $\sum\mathrm{diameter}(B_j)^{m+2}$, which shows that $dim_{Hau}(V)\geq m+2$. %\pause

\begin{thm}
\textcolor{red}{\em (Gromov 1993)}. Let $M$ be a generic subRiemannian manifold of dimension $n$, Hausdorff dimension $Q$, with an $h$-dimensional distribution. Let $k\leq h$ be such that $h-k\geq (n-h)k$. Then $\alpha(M)\leq \frac{n-k}{Q-k}$.\end{thm}

\end{frame}


\section{Curvature pinching}

\begin{frame}

\frametitle{Curvature pinching}

%\begin{defi}
%Say two Riemannian manifolds are {\em quasiisometric} if there exists a map $f$ between them which is biLipschitz in the large, i.e. $\displaystyle d(x,x')\geq 1 \Rightarrow \frac{1}{L}\leq\frac{d(f(x),f(x'))}{d(x,x')}\leq L$.
%\end{defi}

\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 quasiisometric to a $\delta$-pinched simply connected Riemannian manifold. %\pause 
\end{defi}

\begin{ex}
Rank one symmetric spaces of noncompact type are hyperbolic spaces over the reals $H^{n}_{\R}$, the complex numbers $H^{m}_{\C}$, the quaternions $H^{m}_{\H}$, and the octonions $H^{2}_{\O}$.

Real hyperbolic space has sectional curvature $-1$. Other rank one symmetric spaces are $-\frac{1}{4}$-pinched. %\pause
\end{ex}

\begin{ques}
Is it true that the optimal pinching of $H^{m}_{\C}$, $H^{m}_{\H}$ ($m\geq 2$) and $H^{2}_{\O}$ is $-\frac{1}{4}$ ?
\end{ques}

\end{frame}

\subsection{Visual metrics}

\begin{frame}

\textbf{Facts}. Let $M$ be a simply connected pinched Riemannian manifold.
\begin{itemize}
  \item The ideal boundary, seen from a point $o$, is a sphere.
  \item It carries a {\em visual metric} $d_{o}$ (see picture for U. Hamenst\"adt's definition).
  \item Different visual metrics $d_{o}$ and $d_{o'}$ are equivalent.
  \item Quasiisometric maps between negatively curved Riemannian manifolds induce {\em quasisymmetric} maps between ideal boundaries.
\end{itemize}

\begin{center}
\includegraphics[width=2.5in]{visual.pdf}
\end{center}

\end{frame}

\begin{frame}
%\frametitle{}

Negatively curved symmetric spaces admit simply transitive groups of isometries of the form $\R\ltimes G$ where $G$ is a Carnot group and $\R$ acts by dilations $\delta_t$.
\begin{center}
\includegraphics[width=2in]{horosphere.pdf}
\end{center}%\pause

The eigenvalues $e^{t/2}$, $e^{t}$ of $\delta_t$ reflect extrema of sectional curvature.

\begin{cor}
If $M$ is a rank one symmetric space, visual metrics on its ideal boundary are equivalent to subRiemannian metrics. If $M=H^{m}_{\C}$, $\partial M=\mathbb{H}^{m-1}\cup\{\infty\}$.
\end{cor}


\end{frame}

\subsection{Curvature versus H\"older homeomorphisms}

\begin{frame}


\begin{prop}
Let $M$ be a simply connected $\delta$-pinched Riemannian manifold. Then visual metrics on the ideal boundary of $M$ are $C^{\alpha}$-H\"older equivalent to the round metric, with $\alpha=\sqrt{-\delta}$.
\end{prop}
Indeed, geodesics from a unit ball to a point come together exponentially fast, with exponents ranging from $\sqrt{-\delta}$ to $1$ (Rauch comparison theorem, 1950's).%\pause

\begin{ques}
Let $G$ be a nonabelian Carnot group. Do there exist quasisymmetricly equivalent distances on $G$ which are locally $C^{\alpha}$-H\"older equivalent to Riemannian metrics, with $\alpha>1/2$ ?
\end{ques}%\pause

$$
\xymatrix{
&*+[F-,]{\textrm{H\"older quasisymmetric}}\ar@<1pt>@{=>}[dl] \ar@<-1pt>@{=>}[dr] &\\
*+[F-,]{\textrm{H\"older}}&&*+[F-,]{\textrm{pinching}}}
$$

\end{frame}

%\end{document}

\subsection{Main theorem}

\begin{frame}

\begin{thm}
The optimal pinching of $H^{2}_{\C}$ is equal to $-\frac{1}{4}$.
\end{thm}%\pause

\begin{scheme}
\begin{itemize}
  \item Define, in a quasiisometry-invariant manner, families $\mathcal{R}_{p}$, $p>1$, of algebras of functions on $4$-dimensional negatively curved manifolds $M$, and, given $u\in\mathcal{R}_{p}$, a vectorsubspace $\mathcal{S}_{p}(u)\subset\mathcal{R}_{p}$.
  \item If $M$ is $\delta$-pinched and $p<2+4\sqrt{-\delta}$, then for every $u$, $\mathcal{S}_{p}(u)$ is a subalgebra of $\mathcal{R}_{p}$.
  \item If $M=H^{2}_{\C}$, for all $p\in(4,8)$, there exists (locally) $u\in\mathcal{R}_{p}$ such that $\mathcal{S}_{p}(u)$ is not a subalgebra of $\mathcal{R}_{p}$.
\end{itemize}
\end{scheme}
$\mathcal{R}_{p}$ can be viewed as a quasisymmetrically invariant function space on the visual boundary of $M$. However, $\mathcal{S}_{p}(u)$ does not seem to be definable directly in terms of the visual boundary only.

\end{frame}

\section{$L^p$-cohomology}

\begin{frame}
\frametitle{\texorpdfstring{$L^p$}{}-cohomology}

\begin{defi}
Let $M$ be a Riemannian manifold. Let $p>1$. $L^p$-{\em cohomology} of $M$ is the cohomology of the complex of $L^p$-differential forms on $M$ whose exterior differentials are $L^p$ as well,
\begin{eqnarray*}
H^{k,p}&=&\textrm{closed  }k \textrm{-forms in }L^p /d((k-1) \textrm{-forms in }L^p ),\\ %\pause
R^{k,p} & = &\textrm{closed  } k \textrm{-forms in }L^p/\textrm{closure of }d((k-1) \textrm{-forms in }L^p )
,\\ 
T^{k,p}
& = & \textrm{closure of }d((k-1) \textrm{-forms in }L^p )/ d((k-1) \textrm{-forms in }L^p ).
\end{eqnarray*}
$R^{k,p}$ is called the {\em reduced cohomology}. $T^{k,p}$ is called the
{\em torsion}.%\pause
\end{defi}

\begin{ex}
The real line $\R$.%\pause
\end{ex}

$H^{0,p}=0$. %\pause

\smallskip

$R^{1,p}= 0$, since every function in $L^p (\R)$ can be approximated in $L^p$ with derivatives of compactly supported
functions. Therefore $H^{1,p}$ is only torsion. %\pause

\smallskip

$T^{1,p}$  {\em is non zero and thus infinite dimensional}. %\pause 
Indeed, the 1-form $\frac{dt}{t}$ (cut off near the origin) is in $L^p$ for all $p>1$ but it is not the differential of a function in $L^p$.


\end{frame}

\subsection{Examples}

\begin{frame}

\frametitle{Example : the real hyperbolic plane \texorpdfstring{$H^2_{\R}$}{}
}

Here $H^{0,p} = 0 = H^{2,p}$ for all $p$. %\pause 

If $p=2$, since the Laplacian on $L^2$ functions is bounded below, $T^{1,2} = 0$. Therefore 
\begin{eqnarray*}
H^{1,2} &=& R^{1,2}\\ %\pause
&=&
\{L^2\textrm{ harmonic 1-forms}\} \\ %\pause
&=& \{\textrm{harmonic functions } h
\textrm{ on } H^2_{\R}\textrm{ with } \nabla h\in L^2\}/\R \,.
\end{eqnarray*}
%\pause
Using conformal invariance, switch from hyperbolic metric to euclidean metric on the disk $D$.
\begin{eqnarray*} 
H^{1,2} & = & \{ \textrm{harmonic functions }h\textrm{ on } D \textrm{
with } \nabla h\in L^2\}/\R\\ %\pause
& = & \{ \textrm{Fourier series } \Sigma
a_n e^{in\theta } \textrm{ with }
  a_0 = 0, \Sigma \vert n\vert \, \vert a_n\vert ^2 < +\infty \},%\pause
\end{eqnarray*} 
which is Sobolev space $H^{1/2}(\R/2\pi\Z)$ mod constants.
\vfill%\pause
More generally, for $p>1$, $T^{1,p} = 0$ and $H^{1,p}$ is equal to the Besov space $B^{1/p}_{p,p}(\R/2\pi\Z)$ mod constants.

\end{frame}

\subsection{Royden algebra}

\begin{frame}
%\frametitle{Royden algebra}

\begin{prop}
Let $M$ be a simply connected negatively curved Riemannian manifold. Functions $u$ on $M$ whose differential belongs to $L^p$ have boundary values $u_{\infty}$ on the visual boundary. The cohomology class  $[du]\in H^{1,p}(M)$ vanishes if and only if $u_{\infty}$ is constant.
\end{prop}%\pause
Indeed, since volume in polar coordinates grows exponentially, and $L^p (e^t \,dt)\subset L^1 (dt)$, the radial derivative belongs to $L^1$, so $u_{\infty}(\theta)=\lim_{t\to\infty}u(\theta,t)$ exists a.e. If $u_{\infty}=0$, Sobolev inequality $\|u\|_{L^p}\leq\|du\|_{L^p}$ applies, and $[du]=0$.%\pause

\medskip

This suggests

\begin{defi}
\textcolor{red}{(Bourdon-Pajot 2004).} For a negatively curved manifold $M$, define the {\em Royden algebra} $\mathcal{R}_p (M)$ as the space of $L^{\infty}$ functions $u$ on $M$ such that $du\in L^p$, modulo $L^p  \cap L^{\infty}$ functions.
\end{defi}

Then $\mathcal{R}_p (M)$ identifies with an algebra of functions on the visual boundary of $M$. If $M$ is a symmetric space, $\mathcal{R}_p (M)$ is a (possibly anisotropic) Besov space.

\end{frame}

\subsection{Subalgebra theorem}

\begin{frame}

\begin{rem}
$L^p$-cohomology is quasiisometry invariant. Wedge product $\alpha$, $\beta\mapsto \alpha\wedge\beta$ induces cup-product $[\alpha]\smile[\beta]:H^{k,p}\times H^{k',p}\to H^{k+k',p/2}$ in a quasiisometry invariant manner.
\end{rem}

\begin{defi}
Let $M$ be a simply connected negatively curved manifold, let $p>2$, let $u\in\mathcal{R}_{p}(M)$. Define
\begin{eqnarray*}
\mathcal{S}_{p}(u)=\{v\in \mathcal{R}_{p}(M)\,|\,[dv]\smile [du]=0\in H^{2,p/2}(M)\}. 
\end{eqnarray*} 
\end{defi}%\pause

\textbf{Remark}: As a function space on the visual boundary, $\mathcal{R}_p$ is a quasisymmetric invariant. Not so clear for $\mathcal{S}_{p}(u)$.

\begin{thm}
If $\mathrm{dim}(M)=4$, $M$ is $\delta$-pinched and $p<2+4\sqrt{-\delta}$, then for all $u\in\mathcal{R}_{p}(M)$, $\mathcal{S}_{p}(u)$ is a subalgebra of $\mathcal{R}_{p}(M)$.
\end{thm}

\end{frame}

\subsection{Boundary values for differential forms}

\begin{frame}

\hbox{\vbox{\hsize=5cm \textcolor{blue}{Step 1}. {\em For $q=p/2$ small, closed $L^q$ $2$-forms admit boundary values. %\pause
}\vglue.5pt Use the radial vectorfield $\xi=\frac{\partial}{\partial r}$ in polar coordinates and its flow $\phi_t$, whose derivative is controlled by sectional curvature. \vglue.1cm 
Use Poincaré's homotopy formula : 

For $\alpha$ a closed $2$-form in $L^q$, \break\vglue.1cm
$\displaystyle \phi_{t}^{*}\alpha=\alpha+d\left(\int_{0}^{t}\phi_{s}^{*}\iota_{\xi}\alpha\,ds\right)
$ \break\vglue.1cm
has a limit as $t\to+\infty$ under the assumptions of the theorem.}
\vtop{\includegraphics[width=2.2in]{polar.pdf}}} %\pause

\bigskip

\textcolor{blue}{Step 2}. {\em Boundary value determines cohomology class}. %\pause

\bigskip

\textcolor{blue}{Step 3}. {\em This implies $\mathcal{S}_{p}(u)$ is a subalgebra}. %\pause

Let $v$, $v'\in \mathcal{S}_{p}(u)$. Then $[dv]\smile [du]$ vanishes if and only if its boundary value $dv_{\infty}\wedge du_{\infty}=0$ a.e. Then $v'_{\infty}dv_{\infty}\wedge du_{\infty}+v_{\infty}dv'_{\infty}\wedge du_{\infty}=0$ a.e., showing that $[d(vv')]\smile[du]=0$, i.e. $vv'\in \mathcal{S}_{p}(u)$.

\end{frame}

\subsection{$L^p$-cohomology of $H^{2}_{\C}$}

\begin{frame}

Now we compute $H^{2,q}(H^{2}_{\C})$ for $2<q=p/2<4$.%\pause

\medskip

\textcolor{blue}{Step 1}. {\em Switch point of view. Use horospherical coordinates. View $H^{2}_{\C}$ as a product $\mathbb{H}^1 \times\R$. Prove a Künneth type theorem}.%\pause

\smallskip

\hbox{\vbox{\hsize=5cm 
For $q\notin\{4/3,2,4\}$, differential forms $\alpha$ on $H^{2}_{\C}$ split into components $\alpha_{+}$ and $\alpha_{+}$ which are contracted (resp. expanded) by $\phi_{t}$. Then
$$
h_t :\alpha\mapsto\int_{0}^{t}\phi_{s}^{*}\iota_{\xi}\alpha_{+}\,ds-\int_{-t}^{0}\phi_{s}^{*}\iota_{\xi}\alpha_{-}\,ds
$$
converges as $t\to +\infty$ to a bounded operator $h$ on $L^q$. $P=1-dh-hd$ retracts the $L^q$ de Rham complex onto a complex $\mathcal{B}$ of differential forms on $\mathbb{H}^1$ with missing components and weakly regular coefficients. 
}
\vtop{\includegraphics[width=2.2in]{horosphere.pdf}}}

\end{frame}

\begin{frame}

\textcolor{blue}{Step 2}. {\em If $2<q<4$, this complex is nonzero in degrees 1 and 2}.

$\mathcal{B}^1$ consists of $1$-forms which are multiples of the left-invariant contact form $\tau$ on $\mathbb{H}^1$. %\pause

\medskip

\textcolor{blue}{Step 3}. {\em If $2<q<4$, vanishing of degree $2$ cohomology classes is characterized by a differential equation.}

If $\alpha\in\mathcal{B}^2$ is a 2-form, then $\alpha\in d\mathcal{B}^1$ if and only if $\alpha$ satisfies the linear differential equation
\begin{eqnarray*}
\alpha=d(\frac{\tau\wedge\alpha}{\tau\wedge d\tau}\tau).
\end{eqnarray*}
If $dv\wedge du$ is a solution, $d(v^2)\wedge du$ is not a solution, unless $dv$ is proportional to $du$. %\pause

\bigskip

\textbf{Failure of the subalgebra theorem for $H^{2}_{\C}$}.

In coordinates $(x,y,z)$ on $\mathbb{H}^1$, one can take (locally) $u=y$ and  $v=x$. Then $dv\wedge du=-d\tau$ belongs to $d\mathcal{B}^1$, whereas $d(v^2)\wedge du$ does not. So for $4<p=2q<8$, $\mathcal{S}_{p}(u)$ is not (locally) a subalgebra of $\mathcal{R}_{p}(H^{2}_{\C})$.%\pause

\bigskip

\textbf{Other rank one symmetric spaces}.

The comparison theorem works for all of them: in the definition of $\mathcal{S}_{\kappa}$, replace $du$ by a cohomology class $\kappa$ of degree $1$, resp. $3$ resp. $7$. Steps 1 and 2 of the $L^q$ computation in degree $2$ resp. $4$ resp. $8$ are unchanged. It turns out that for all spaces but $H^{2}_{\C}$, the differential equation of Step 3 is a consequence of $d\alpha=0$, so $\mathcal{S}_{\kappa}$ is an algebra in these cases.

\end{frame}


\begin{frame}
\frametitle{Questions from the audience}

\begin{itemize}
  \item \textbf{Peter Ha\"issinski}: {\em doesn't conformal dimension provide a lower bound on pinching~?}

Answer: It does. If $M$ is $n$-dimensional and $\delta$-pinched, then conformal dimension is $\leq (n-1)/\sqrt{-\delta}$. This yields the bound $\delta(H^{2}_{\C})\leq -9/16$. Also, this gives a partial (non sharp) answer to the H\"older+quasisymmetric problem for Heisenberg group: if there is a map $\R^3 \to \mathbb{H}^1$ which is the composition of a $C^{\alpha}$ (local) homeomorphism and a quasisymmetric (local) homeomorphism, then $\alpha\geq 3/4$.
  \item \textbf{Mario Bonk}: {\em how do cup-products behave for Rickmann's rug and the corresponding negatively curved Riemannian homogeneous space ?}
  
Answer: Consider the $3$-dimensional rug leading to the solvable group $S=\R\ltimes\R^3$ with $\R$ acting on $\R^3$ via the diagonal matrix $diag(e^{t/2},e^{t/2},e^{t})$. It is $-\frac{1}{4}$-pinched. For $2<q<4$, any closed form in $\mathcal{B}^2$ is exact in a neighborhood of a point of the ideal boundary. So the local version of $\mathcal{S}_{u}$ is an algebra. However,
$H^{2,q}(S)$ has torsion, indicating that the condition for a closed form in $\mathcal{B}^2$ to be globally exact is subtle, so the global version of $\mathcal{S}_{u}$ is probably not an algebra. Nevertheless, the fact that $T^{2,q}(S)\not=0$ alone suffices to prove that $\delta(S)=-\frac{1}{4}$.
\end{itemize}

\end{frame}


\end{document}