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\title{Flat compactness of normal Rumin currents on the Heisenberg group}
\author{Pierre Pansu, joint with Antoine Julia, Universit\'e Paris-Saclay}
\date{June 30th, 2022}

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\begin{document}

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

\begin{frame}

\begin{center}
\textbf{A few milestones in Bruno Franchi and Raul Serapioni's joint achievements}
\end{center}

\begin{itemize}

  \item 2001: Franchi, Serapioni and Serra Cassano inaugurate \emph{Geometric Measure Theory} on Carnot groups. Their first rectifiability result is crucial in Cheeger, Kleiner and Naor's work on the Goemans-Linial conjecture.\pause
  \item 2005: Franchi, Serapioni and Serra Cassano introduce intrinsic Lipschitz graphs, as candidates for models of rectifiable sets. These play a key role in Naor and Young's final answer to Goemans-Linial's conjecture.\pause
  \item 2005-2016: Franchi and Serapioni (often with Serra-Cassano) study the differentiability of intrinsic Lipschitz functions, guided by the idea of currents.
\end{itemize}
\end{frame}

\section{Rumin currents}

\subsection{Results}

\begin{frame}

We are interested in Rumin currents on Heisenberg groups, defined by duality from Rumin differential forms.\pause

\medskip

We are yet unable to chose the appropriate definition of integral Rumin currents and attack the question of their rectifiability (Franchi and Serapioni's dream).\pause

\medskip

Meanwhile, we collect soft results on normal Rumin currents.\pause

\begin{thm}[Compactness]
The space of normal Rumin currents with bounded norm and support in a fixed compact subset of the Heisenberg group is compact in flat topology.
\end{thm}\pause

\begin{thm}[Representation of charges]
A (certain) dual of the space of normal Rumin currents with compact support is the space $C^0+d_R C^0$.
\end{thm}

This is an avatar of a result by T. De Pauw, L. Moonens and W. Pfeffer (2009) in Euclidean space.

\end{frame}

\subsection{Definitions}

\begin{frame}

Rumin forms on Heisenberg group can be viewed as a subset of differential forms, defined by the vanishing of certain components. M. Rumin has defined a differential $d_R$ on Rumin forms that satisfies $d_R\circ d_R=0$ and whose cohomology vanishes locally. $d_R$ has order two in the middle dimension.\pause

\medskip

A \emph{Rumin current} is a continuous functional on the space of smooth compactly supported Rumin forms. \pause The boundary operator $\partial_R$ is defined by duality, 
$$
\langle \partial_R T,\omega \rangle =\langle T,d_R\omega \rangle.
$$\pause 
The \emph{support} of $T$ is such that $T$ vanishes on forms with support in the complement.\pause

\medskip

The \emph{mass} $M(T)$ of a Rumin current is
$$
M(T)=\sup\{\langle T,\omega \rangle\,;\, \|\omega\|_\infty \le 1\}.
$$

\medskip

A Rumin current is \emph{normal} if $N(T):=\max\{M(T),M(\partial_R T)\}<+\infty$.\pause

\medskip

The \emph{flat norm} of a Rumin current $T$ is
$$
F(T):=\min\{M(R)+M(S)\,;\, T=R+\partial_R S\}.
$$

\end{frame}

\subsection{Examples}

\begin{frame}

\textbf{Diffuse currents}. A $C^1$ Rumin form $\phi$ defines a current $P(\phi)$ of complementary dimension by
$$
\langle P(\phi),\omega \rangle = \int\phi\wedge\omega.
$$\pause 
Then $M(P(\phi))=\|\phi\|_1$, $\partial_R P(\phi)=\pm P(d_R \phi)$, $N(P(\phi))=\max\{\|\phi\|_1,\|d_R\phi\|_1\}$. \pause

Currents $P(\phi)$ are dense in flat norm in the space of normal Rumin currents. \pause

\bigskip

\textbf{Currents of integration}. In $\mathbb{H}^1$, a smooth 2-submanifold $V$ with boundary defines a Rumin current $T_V$ of mass equal to its Hausdorff $3$-dimensional measure. \pause

\medskip

For a smooth $1$-manifold $C$ with boundary, in order to have $\partial_R T_V=T_{\partial V}$, one must define
$$
\langle T_C,\omega \rangle = \int_C \omega-\frac{d\omega}{d\theta}\theta .
$$\pause
(Indeed, $\omega-\frac{d\omega}{d\theta}\theta$ depends only on the horizontal part of $\omega$ and $d_R\omega=d(\omega-\frac{d\omega}{d\theta}\theta)$).\pause 

\medskip

Therefore $M(T_C)<\infty$ only if $C$ is horizontal. In that case, $M(T_C)$ equals Hausdorff $1$-dimensional measure.

\end{frame}

\section{Compactness}

\subsection{Duality}

\begin{frame}

\begin{thm}[Compactness]
The space of normal Rumin currents with bounded norm and support in a fixed compact subset of the Heisenberg group is compact in flat topology.
\end{thm}\pause

By duality (the adjoint of a compact operator is compact), this follows from a corresponding statement for Rumin forms. Consider the Heisenberg nilmanifold $Z=\mathbb{H}^n/\mathbb{H}^n_\Z$.\pause

\begin{prop}
  Let $\mathcal{X}$ be the space of Rumin forms on $Z$ with vanishing averages, equipped with the norm
        \[
           \|\omega\|_\mathcal{X}:=\max\{ |\omega|_\infty , |d_R \omega |_\infty\},
        \]
    and let $\mathcal{Y}$ be the space of Rumin forms on $Z$ equipped with the norm
        \[
            \|\omega\|_\mathcal{Y}:= \inf \{ |\phi|_\infty + |\psi|_\infty\,;\, \omega= \psi+d_R \phi \}.
        \]
    Then the injection $\mathcal{X}\hookrightarrow \mathcal{Y}$ is compact.
\end{prop}

\end{frame}

\subsection{Primitives of $L^\infty$ Rumin forms}

\begin{frame}

Pick $p>2n+2$. Then every closed Rumin form $\omega\in L^q$ has a primitive $\phi\in W^{1,q}$. For instance, $\phi=d_R^*\Delta_R^{-1}\omega$ (slightly different formula in middle dimension). \pause 

\medskip
When passing to $Z$, a parametrix $P$ for the Rumin Laplacian $\Delta_R$ yields an operator $Q=d_R^*P$ mapping $L^p$ to $W^{1,p}$ such that
$$
S=1-d_R Q-Qd_R
$$
is a smoothing operator (Baldi-Franchi-Pansu).\pause

\medskip

If $(\omega_j)$ is bounded in $\mathcal{X}$, 
$$
\omega_j=S\omega_j+d_R(Q\omega_j)+Q(d_R\omega_j)
$$
where all summands are bounded in $W^{1,p}$. \pause

\medskip

Sobolev's embedding $W^{1,p}\hookrightarrow C^{\alpha}$, $\alpha=1-(2n+2)/p$, implies that each summand is bounded in $C^\alpha$, hence compact in $L^\infty$. 

\medskip

Thus $(\omega_j)$ is compact in $\mathcal{Y}$.

\end{frame}

\section{Charges}

\subsection{Definition}

\begin{frame}

For $K$ be a compact subset of Heisenberg group and $\nu>0$, let $S(K,\nu)$ denote the space of Rumin currents $T$ with support in $K$ such that $N(T)\le\nu$, equipped with the flat norm. This is a compact convex set.\pause


\begin{defi}
A \textbf{Rumin charge} is a linear functional on the space of compactly supported normal Rumin currents, which is continuous on each $S(K,\nu)$.
\end{defi}
\pause

\begin{thm}[Representation of Rumin charges]
The space of Rumin charges is $C^0+dC^0$, i.e. if $\phi$ is a continuous Rumin form on Heisenberg group, both $\phi$ and $d_R\phi$ define Rumin charges. Conversely, every Rumin charge is of the form $\phi+d_R\psi$, where $\phi$ and $\psi$ are continuous Rumin forms on Heisenberg group.
\end{thm}\pause

The Euclidean case is part of Laurent Moonens' PhD Thesis, UC Louvain, 2008.

\end{frame}

\subsection{Semireflexivity}

\begin{frame}

Fix a compact set $K$, let $\mathcal{M}_K$ (resp. $\mathcal{N}_K$) be the space of currents of finite mass (resp. finite $N$ norm) with support in $K$. Then $\mathcal{M}_K = C^0(K)'$, 
$$
\mathcal{N}_K=\mathcal{M}_K \cap d_R^{-1}(\mathcal{M}_K)=(C^0(K)+d_R C^0(K))'.
$$
So we are claiming reflexivity of $C^0+d_R C^0$, which fails! \pause The point is the change of topology on $\mathcal{N}_K$, passing from the $N$ norm to the flat norm. The keyword is \emph{semireflexivity}. \pause


\begin{defi}[Bourbaki]
A linear form on a topological vectorspace is \textbf{strongly continuous} if it is bounded on bounded subsets.

A locally convex topological vectorspace $\mathcal{X}$ is \textbf{semireflexive} if every strongly continuous linear form on its topological dual $\mathcal{X}'$ arises from an element of $\mathcal{X}$.
\end{defi}
%\pause

%Here is a semireflexivity criterion: \emph{$X$ is semireflexive if and only if every bounded subset of $X$ is weakly relatively compact.}

\end{frame}

\subsection{Proof of the representation of charges}

\begin{frame}

Let $\mathcal{X}$ be a locally convex space, let $\mathcal{S}$ be a dilation stable family of convex subsets of $\mathcal{X}$. There is a topology $\mathfrak{T}_{\mathcal{S}}$ on $\mathcal{X}$, inducing the initial topology on each $S\in\mathcal{S}$, such that a linear map $\mathcal{X}\to \mathcal{Y}$ is continuous if and only if all its restrictions to elements of $\mathcal{S}$ are continuous.\pause

\begin{thm}[T. De Pauw, L. Moonens, W. Pfeffer 2009]
Let $\mathcal{X}$ be a locally convex space, let $\mathcal{S}$ be an exhausting family of \textbf{compact} convex subsets of $X$. Then $(X,\mathfrak{T}_{\mathcal{S}})$ is semireflexive.
\end{thm}\pause

If $\mathcal{N}$ is the space of compactly supported normal Rumin currents, endowed with the flat topology, and $\mathcal{S}=\{S_{K,\nu}\}$, then Rumin charges are precisely continuous linear functionals on $(\mathcal{N},\mathfrak{T}_{\mathcal{S}})$,
$$
\mathcal{CH}\simeq(\mathcal{N},\mathfrak{T}_{S})'.
$$
\pause

The theorem applies. By semireflexivity, every strongly continuous linear functional on the space of charges $\mathcal{CH}$ arises from a normal current of compact support,
$$
(\mathcal{N},\mathfrak{T}_{\mathcal{S}})\simeq \mathcal{CH}^*.
$$

\end{frame}

\begin{frame}

The second ingredient is the identification of bounded subsets of $C^0$.

\begin{prop}
If a subset of $\mathcal{N}$ is uniformly strongly bounded as linear functionals on $C^0$, then all its elements have support in the same compact set of Heisenberg group. 
\end{prop}\pause

We want to show that $\Theta:C^0\oplus C^0\to \mathcal{CH}$, $(\phi,\psi)\mapsto \phi+d_R\psi$ is onto. We proceed by showing that its adjoint $\Theta^*:\mathcal{N}=\mathcal{CH}^*\to (C^0\oplus C^0)^*$, given by
$$
\langle \Theta^*(T),(\phi,\psi) \rangle = \langle T,\phi+d_R\psi \rangle=\langle T,\phi \rangle+\langle \partial_R T,\psi \rangle,
$$
is proper. \pause If $S\subset \mathcal{N}$ and $\Theta^*(S)$ is bounded on all bounded subsets of $(C^0\oplus C^0)^*$, then, for $T\in S$, $N(T)$ is bounded and $\mathrm{supp}(T)$ is in a common compact set. \pause So $S$ is flat-compact. \pause

\medskip

Hence the range of $\Theta^*$ is closed. This implies that $\Theta$ is onto, i.e. every charge can be written $\phi+d_R\psi$, for $\phi,\psi\in C^0$.

\end{frame}





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