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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{boxes}}

\begin{document}


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

\begin{frame}

\frametitle{What is $L^p$-cohomology ?}
\section{What is $L^p$-cohomology ?}

\begin{eqnarray*}
topological~space\quad&\to&\quad cohomology\\
manifold\quad&\to&\quad de~Rham~cohomology\\\pause
metric~space\quad&\to&\quad cohomology~with~decay~condition\\
Riemannian~manifold\quad&\to&\quad de~Rham~cohomology~with~decay~condition\pause
\end{eqnarray*}


\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}.
\end{defi}

\end{frame}
\begin{frame}

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

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}
\begin{frame}


\frametitle{Example : the real line $\R$}
\section{Example : the real line $\R$}

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

\bigskip

$R^{1,p}= 0$, \pause 

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

\bigskip

$T^{1,p}$  {\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}
\begin{frame}

\frametitle{What are our favourite spaces ?}
\section{What are our favourite spaces ?}

\begin{itemize}
  \item $L^p$-cohomology has been used (L. Saper, S. Zucker) to study manifolds with \textcolor{blue}{thin ends}, e.g. locally symmetric spaces of finite volume. The answer is related to the topology of a compactification.

\bigskip

  \item In this talk : manifolds with \textcolor{blue}{large ends}, e.g. symmetric spaces themselves. $L^p$-cohomology is related to analytic features of a compactification.
\end{itemize} \pause

\vskip1cm

In conclusion,

\begin{itemize}
  \item $L^p$-cohomology tastes like harmonic analysis.
  \item We shall apply it to a problem in Riemannian geometry.
  \item Ideas from algebraic topology play a role.
\end{itemize}

\end{frame}

\begin{frame}

\frametitle{Curvature pinching}
\section{Curvature pinching}

\begin{rem}
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 $-\frac{1}{4}$-{\em pinched}, i.e. their sectional curvature ranges between $-1$ and $-\frac{1}{4}$. \pause
\end{rem}

\begin{defi}
Define the {\em optimal pinching} $\delta(M)$ of a Riemannian manifold $M$ as the least $\delta\geq-1$ such that $G$ is bi-Lipschitz equivalent to a $\delta$-pinched Riemannian manifold. \pause 
\end{defi}

\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}
\begin{frame}

\frametitle{$L^p$-cohomology of real hyperbolic spaces}
\section{$L^p$-cohomology of real hyperbolic spaces}

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

\end{frame}
\begin{frame}

\frametitle{$L^p$-torsion and pinching}
\section{$L^p$-torsion and pinching}

\begin{thm}
If $M^{n}$ is simply connected and $\delta$-pinched for some $\delta\in[-1,0)$, then
\begin{eqnarray*}
p<1+\frac{n-k}{k-1}\sqrt{-\delta} \quad\Rightarrow\quad T^{k,p}(M)=0.\pause
\end{eqnarray*}
\end{thm}
This is sharp. For instance, consider the semidirect product $G=\R^3 \rtimes_{\alpha}\R$ where $\alpha=diag(1,1,2)$. 
\begin{itemize}
  \item It admits a $-\frac{1}{4}$-pinched left-invariant Riemannian metric, therefore $\delta(G)\leq -\frac{1}{4}$.
  \item It has $T^{2,p}(G)\not=0$ for $2<p\leq 4$. This implies that $\delta(G)=-\frac{1}{4}$.\pause
\end{itemize} 

\begin{rem}
Complex hyperbolic plane $H^{2}_{\C}$ is isometric to $G'=Heis^3 \rtimes_{\alpha}\R$ where $\alpha=diag(1,1,2)$ and $Heis$ denotes the Heisenberg group. Therefore it is very close to $G$.\pause
\end{rem}

\begin{thm}
$T^{2,p}(H^{2}_{\C})=0$ for $2<p< 4$.
\end{thm}


\end{frame}

\begin{frame}

\frametitle{Proof of torsion comparison theorem}
\section{Proof of torsion comparison theorem}

\hbox{\vbox{\hsize=5cm \textcolor{blue}{Step 1}. {\em For $p$ large, closed $L^p$ 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 $k$-form in $L^p$, \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

\noindent This boundary value map injects $H^{k,p}$ into a function space of closed forms on the ideal boundary, showing that $H^{k,p}$ is Hausdorff.

\end{frame}

\begin{frame}

\frametitle{Proof of torsion vanishing for $H^{2}_{\C}$}
\section{Proof of torsion vanishing for $H^{2}_{\C}$}

Switch point of view. Use horospherical coordinates. 
\vglue1.05cm 

\hskip5cm\includegraphics[width=2.2in]{horosphere.pdf}
\vglue 6cm

\end{frame}

\begin{frame}

\frametitle{Proof of torsion vanishing for $H^{2}_{\C}$}
\section{Proof of torsion vanishing for $H^{2}_{\C}$}

Switch point of view. Use horospherical coordinates. View $H^{2}_{\C}$ as a product $Heis\times\R$. Prove a Künneth type theorem.
\vglue.8cm 

\hskip5cm\includegraphics[width=2.2in]{horosphere.pdf}
\vglue 6cm

\end{frame}

\begin{frame}

\frametitle{Proof of torsion vanishing for $H^{2}_{\C}$}
\section{Proof of torsion vanishing for $H^{2}_{\C}$}

Switch point of view. Use horospherical coordinates.
View $H^{2}_{\C}$ as a product $Heis\times\R$. Prove a Künneth type theorem.\vglue.5pt 
\hbox{\vbox{\hsize=5cm 
\textcolor{blue}{Step 1}. For $p\notin\{4/3,2,4\}$, 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 $L^p$. $P=1-dh-hd$ retracts the $L^p$ de Rham complex onto a complex $\mathcal{B}$ of differential forms on $Heis^3$ with missing components and weakly regular coefficients. 
}
\vtop{\includegraphics[width=2.2in]{horosphere.pdf}}} \pause


\medskip

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

Let $\tau$ denote the invariant contact form on $Heis^3$. Then $\mathcal{B}^1$ consists of 1-forms which are multiples of $\tau$. Since
\begin{eqnarray*}
d(f\tau)=df\wedge\tau+f\,d\tau,
\end{eqnarray*}
$d(f\tau)$ determines $f$, therefore $d:\mathcal{B}^1 \to \mathcal{B}^2$ has closed range.

\end{frame}

\begin{frame}

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

\begin{center}
\includegraphics[width=2.5in]{table0.pdf}
\end{center}

\end{frame}


\begin{frame}

\frametitle{Subalgebra theorem}
\section{Subalgebra theorem}

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

\medskip

Obviously, $u\in \mathcal{R}_p (M)$ implies $[du]\in H^{1,p}(M)$. \pause

Recall that cup-product $\smile$ is well defined : $H^{k,q}(M)\times H^{\ell,r}(M)\to H^{k+\ell,p}(M)$ provided $\frac{1}{p}=\frac{1}{q}+\frac{1}{r}$. \pause

\begin{thm}
{\em (Work in progress)}.
Assume $M^{n}$ is simply connected and $\delta$-pinched for some $\delta\in(-1,0)$. Assume 
\begin{eqnarray*}
p<1+\frac{n-k}{k-1}\sqrt{-\delta},\quad r<1+\frac{n-k+1}{k-2}\sqrt{-\delta}\quad \textrm{and}\quad \frac{1}{p}=\frac{1}{q}+\frac{1}{r}.
\end{eqnarray*}
Pick a class $\kappa\in H^{k-1,r}(M)$. Then the set of $u\in\mathcal{R}_q (M)$ such that $[du]\smile\kappa=0$ in $H^{k,p}(M)$ is a subalgebra.
\end{thm}

\begin{ex}
$n=4$, $k=2$, $q=r=2p$. For $\delta<-\frac{1}{4}$, the theorem applies for some $p>2$.
\end{ex}

\end{frame}


\begin{frame}

\frametitle{Failure of subalgebra theorem for $H^2_{\C}$}
\section{Failure of subalgebra theorem for $H^2_{\C}$}

\begin{prop}
{\em (Work in progress)}.
Let $2<p<4$. There exists a conical open set $U\subset H^2_{\C}$, a class $\kappa\in H^{1,2p}(U)$ and a function $u\in\mathcal{R}_{2p} (U)$ such that $[du]\smile\kappa=0$ in $H^{2,p}(U)$ but $[d(u^2)]\smile\kappa\not=0$ in $H^{2,p}(U)$.
\end{prop} \pause

\begin{rem}
This should imply that the optimal pinching for $H^2_{\C}$ is $-\frac{1}{4}$, provided subalgebra theorem localizes on conical open sets.
\end{rem} \pause

\proof

If $\alpha\in\mathcal{B}^2$ is a 2-form, then $\alpha\in d\mathcal{B}^1$ implies that there exists a function $f$ such that $\alpha=df\wedge\tau+f\,d\tau$. Then $\displaystyle f=\frac{\tau\wedge\alpha}{\tau\wedge d\tau}$. Thus $\alpha$ satisfies a linear differential equation
\begin{eqnarray*}
\alpha=d(\frac{\tau\wedge\alpha}{\tau\wedge d\tau}\tau).
\end{eqnarray*}
If $du\wedge\beta$ is a solution, $d(u^2)\wedge\beta$ is not a solution, unless $\beta$ is proportional to $du$. \pause

\begin{ex}
In coordinates, if $\tau=dz-x\,dy$, one can take $u=x$, $\beta=dy$.
\end{ex}

\end{frame}

\end{document}

\begin{frame}

\frametitle{Non-vanishing of torsion}
\section{Non-vanishing of torsion}

Use Poincaré duality. Let $p'=p/p-1$ denote the conjugate exponent. In order to prove that a closed $k$-form $\alpha$ is nonzero in cohomology, it suffices to construct a sequence $\psi_{j}$ of $(n-k)$-forms such that $\n{d\psi_j}_{L^{p'}}$ tends to zero but $\int\alpha\wedge\psi_{j}$ does not tend to zero.\pause

\bigskip

In conclusion, we have used
\begin{itemize}
  \item Poincaré duality.
  \item A deformation retraction of space onto a subspace, with controlled effect on the $L^p$-norms of forms. For certain ranges of $p$, this provides a boundary value.\pause
\end{itemize}

\bigskip

\begin{ques}
\begin{itemize}
  \item For rank 1 symmetric spaces, $T^{k,p}=0$ except for at most 1 value of $p$ in each degree.
  \item For higher rank symmetric spaces, $H^{k,p}=0$ for $k<rank$, $T^{k,p}=0$ for $k=rank$.
  \item For $k=rank$, $R^{k,p}\not=0$ for $p$ large, and $R^{k,p}$ is a function space on the maximal boundary.
  \item For each $p>1$, there exists $k$ such that $H^{k,p}\not=0$.
\end{itemize}
\end{ques}

\end{frame}

\begin{frame}

\frametitle{Functoriality}
\section{Functoriality}

\begin{eqnarray*}
\textrm{cohomology}&\to&\textrm{continuous maps}\\\pause
L^p \textrm{-cohomology}&\to&\textrm{uniform maps}.\pause
\end{eqnarray*}

\begin{defi}
A map $f:X\to Y$ between metric spaces is {\em uniform} if $d(f(x),f(x'))$ is bounded from above in terms of $d(x,x')$ only. \pause
\end{defi}

\begin{ex}
The obvious map $\Z\to \R$ is uniform. Any homomorphism between groups (with left invariant metrics) is uniform. The parametrization of a cusp by a punctured disk is not uniform.\pause
\end{ex}

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

\begin{prop}
Among contractible Riemannian manifolds admitting a cocompact isometric group action, $L^p$-cohomology is natural under uniform maps.
\end{prop}


\end{frame}

\begin{frame}

\frametitle{$L^p$
-cohomology of $G$}
\section{$L^p$-cohomology of $G$}

\begin{center}
\includegraphics[width=3in]{table1.pdf}
\end{center}

\end{frame}



