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

\title{$L^p$-cohomology and curvature pinching}
\author{P. Pansu}
\date{August 25th, 2011}

\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{fac}[lem]{Fact}
\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}


\begin{frame}

Goal: explain how a tool from quantitative topology ($L^p$-cohomology) helps in solving a question in Riemannian geometry (optimal curvature pinching).

\end{frame}


\section{\texorpdfstring{$L^p$}{}-cohomology}

\begin{frame}

\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{rem}
If $M$ is compact, $L^p$-cohomology equals cohomology.%\pause
\end{rem}

\begin{rem}
For (uniformly) contractible spaces, $L^p$-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}

\end{frame}

\subsection{Examples}

\begin{frame}

\begin{defi}
$H^{k,p}=$ closed  $k$-forms in $L^p /d((k-1)-$forms in $L^p )$,

$R^{k,p}=$ closed  $k$-forms in $L^p/$ closure of $d((k-1)$-forms in $L^p )$,

$T^{k,p}=$ closure of $d((k-1)$-forms in $L^p )/ d((k-1)$-forms in $L^p )$.
\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}

\begin{frame}

\begin{defi}
$H^{k,p}=$ closed  $k$-forms in $L^p /d((k-1)-$forms in $L^p )$,

$R^{k,p}=$ closed  $k$-forms in $L^p/$ closure of $d((k-1)$-forms in $L^p )$,

$T^{k,p}=$ closure of $d((k-1)$-forms in $L^p )/ d((k-1)$-forms in $L^p )$.
\end{defi}

\begin{ex}
The real hyperbolic plane $H^2_{\R}$.%\pause
\end{ex}

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.

%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{Boundary values: \texorpdfstring{$1$}{}-forms}

\begin{frame}

\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 homogeneous, $\mathcal{R}_p (M)$ is a (possibly anisotropic) Besov space.

\end{frame}

\subsection{Boundary values: higher degrees}

\begin{frame}

\hbox{\vbox{\hsize=5cm \textcolor{blue}{Step 1}. {\em For $q$ 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 some curvature pinching assumption.}
\vtop{\includegraphics[width=2.2in]{polar.pdf}}} %\pause

\bigskip

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

\begin{thm}
If $\mathrm{dim}(M)=4$, $M$ is $\delta$-pinched and $q<1+2\sqrt{-\delta}$, then a boundary value operator is defined, it injects $H^{2,q}$ into closed forms on the boundary. In particular, $T^{2,q}=0$.
\end{thm}
$\delta$-pinched means sectional curvature $\in[-1,\delta]$.

\end{frame}


\section{Curvature pinching}

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

\begin{ex}
Let $M=\R^4$ with metric $dt^2 +e^{t}dx^{2}+e^{t}dy^{2}+e^{2t}dz^{2}$. Then $\delta(M)=-\frac{1}{4}$.%\pause
\end{ex}

\proof.
$M$ is $-\frac{1}{4}$-pinched. $T^{2,q}(M)\not=0$ for $2<q<4$.%\pause

\begin{thm}
\textcolor{red}{If $\mathrm{dim}(M)=4$, $M$ is $\delta$-pinched and $q<1+2\sqrt{-\delta}$, then $T^{2,q}=0$}.
\end{thm}

\begin{rem}
\textcolor{red}{For (uniformly) contractible spaces, $L^p$-cohomology is quasiisometry invariant.}
\end{rem}

\end{frame}

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

\begin{ex}
Let $M=\R^4$ with metric $dt^2 +e^{t}dx^{2}+e^{t}dy^{2}+e^{2t}dz^{2}$. Then $\delta(M)=-\frac{1}{4}$.
\end{ex}
$M$ is isometric to a left-invariant metric on the Lie group $\R\ltimes\R^3$. %\pause

\begin{ex}
Let $M=\R^4$ with metric $dt^2 +e^{t}dx^{2}+e^{t}dy^{2}+e^{2t}(dz-xdy)^{2}$. Then $M$ is $-\frac{1}{4}$-pinched, but $T^{2,q}(M)=0$ for $2\leq q<4$.%\pause
\end{ex}

$M$ is isometric to a left-invariant metric on the Lie group $\R\ltimes Heis$.
%\pause

$M$ is isometric to {\em complex hyperbolic plane} $H^{2}_{\C}$.%\pause

\end{frame}

\subsection{Rank one symmetric spaces}

\begin{frame}

Complex hyperbolic space $H^{m}_{\C}$ is a metric on the ball in $\C^m$ which is invariant under all holomorphic automorphisms. In $H^{m}_{\C}$, every geodesic is contained in a complex line (a totally geodesic plane of curvature $-1$). Every plane orthogonal to a complex line integrates into a totally geodesic plane of curvature $-\frac{1}{4}$. %\pause

Spheres in $H^{m}_{\C}$ are homogeneous under conjugates of $U(m)$. Horospheres are homogeneous under Heisenberg group $Heis^{m-1}$. %\pause

There are quaternionic $H^{m}_{\H}$ and octonionic $H^{2}_{\O}$ siblings.%\pause 

All are $-\frac{1}{4}$-pinched.%\pause

Together with ordinary hyperbolic space $H^{n}_{\R}$, these constitute the list of all negatively curved symmetric spaces (i.e. geodesic inversion is an isometry), \textcolor{red}{E. Cartan, 1925}.%\pause 

And also the list of all noncompact Riemannian manifolds with $2$-point transitive isometry groups, \textcolor{red}{J. Tits, 1955}. %\pause 

All have compact quotients.

\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{The periodic version}

\begin{frame}

If one sticks to Riemannian manifolds admitting a compact quotient, the answer has been known since the 1980's, including the equality case.

\begin{fac}
Let $N$ be a compact quotient of $H^{m}_{\C}$, $H^{m}_{\H}$ ($m\geq 2$) or $H^{2}_{\O}$. If a metric on $N$ is $-\frac{1}{4}$-pinched, then it lifts to a symmetric metric.
\end{fac}

This is due to 
\begin{itemize}
  \item \textcolor{red}{M. Ville, 1984} for $H^{2}_{\C}$ (estimate on a characteristic class),  
  \item \textcolor{red}{L. Hern{\'a}ndez-Lamoneda, 1991}, and independently \textcolor{red}{S.T. Yau and F. Zheng, 1991} for $H^{m}_{\C}$,
  \item \textcolor{red}{N. Mok, Y.T. Siu and S.K. Yeung, 1993}, and independently \textcolor{red}{J. Jost and S.T. Yau, 1993} for other spaces (harmonic maps).
\end{itemize}

\end{frame}


\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 Recall Royden algebras $\mathcal{R}_{p}(M)$, $p>1$, are quasi-isometry invariants.
  \item Given $u\in\mathcal{R}_{p}$, define a vectorsubspace $\mathcal{S}_{p}(u)\subset\mathcal{R}_{p}$, in a quasi-isometry invariant manner.
  \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}%\pause

\begin{defi}
Let $M$ be a simply connected negatively curved manifold, let $p>4$, 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}

\end{frame}

\section{Proofs}

\subsection{Proof of subalgebra theorem}

\begin{frame}

\begin{defi}
Let $M$ be a simply connected negatively curved manifold, let $p>4$, 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}

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

\proof. 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{\texorpdfstring{$L^p$}{}-cohomology of \texorpdfstring{$H^{2}_{\C}$}{}
}

\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}

\end{document}