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\usetheme{Warsaw}
\title[CNAA Genova 2012: \it Distance to curves and surfaces\rm]{Distance to curves and surfaces in the Heisenberg group}
\author{Nicola Arcozzi}
\institute{Universit\`a di Bologna}
\date{3 giugno 2012}
\begin{document}

\begin{frame}
\titlepage
\end{frame}
\begin{frame}{Survey based on...}
\begin{itemize}
 \item Fausto Ferrari, N.A. \it Metric normal and distance function in the Heisenberg group, \rm Math.Z. 2007. 
 \item Fausto Ferrari, N.A. \it The Hessian of the distance from a surface in the Heisenberg group, \rm Ann. Acad. Fenn. 2008. 
 \item Annalisa Baldi, N.A. \it From Gruschin to Heisenberg via an isoperimetric problem, \rm JMAA 2008. 
 \item N.A., Fausto Ferrari, Francescopaolo Montefalcone \it CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian Carnot groups , \rm preprint 2009. 
 \item N.A. \it Distance to a curve in the Heisenberg group, \rm in preparation 2012. 
 \item Fausto Ferrari, N.A. \it A variational approximation of the perimeter with second order
  penalization in the Heisenberg group, \rm in preparation 2012. 
\end{itemize}
\end{frame}
\begin{frame}{Heisenberg group and CC geometry}
 \begin{itemize}
  \item Group: $\HH=\RR^3\ni(x,y,t)$, $(x_1,y_1,t_1)\cdot(x_2,y_2,t_2)=(x_1+x_2,y_1+y_2,t_1+t_2+1/2(x_1y_2-y_1x_2)).$
  \item Lie algebra: $X=\partial_x-\frac{y}{2}\partial_t$, $Y=\partial_y+\frac{x}{2}\partial_t$, $T=[X,Y]=\partial_t$: $\hfrak=\text{span}\{X,Y,T\}$.
  \item Stratification: $H=V_1=\text\{X,Y\}$, $V=V_2=[V_1,V_2]=\text{span}\{T\}$.
  \item The CC length of a curve: $\gamma:[a,b]\to\HH$, $\dot{\gamma}=\alpha X+\beta Y+ mT$ is   
\begin{block}{CC length}
$\text{length}(\gamma)=\int_a^b\sqrt{\alpha(\tau)^2+\beta(\tau)^2+\infty^2\cdot m(\tau^2)}d\tau.$
\end{block}
  \item $\gamma$ is \it horizontal \rm if $m\equiv0$ (iff $\text{length}(\gamma)<\infty$, iff $\dot{\gamma}\in H$).
  \item $d(P,Q)=\inf\left\{\text{length}(\gamma):\ \gamma(a)=P,\ \gamma(b)=Q\right\}.$
  \item $d$ is a distance on $\HH$, realized by the length of \it geodesics.\rm
  \item $d(O,(x,y,t))\approx\left((x^2+y^2)^2+t^2\right)^{1/2}$.
  \item $\gamma=(x,y,t)$ is horizontal iff $dt=\frac{xdy-ydx}{2}$: we can give an interpretation of $\text{length}$ in terms of areas.
  \end{itemize}
\end{frame}
\begin{frame}{$\Delta t=\int_{\pi(\gamma)}\frac{xdy-ydx}{2}=\iint_{\text{int}(\pi(\gamma))}dxdx=\text{Area}(\text{int}(\pi(\gamma)))$.}
\begin{center}
 \includegraphics[angle=90,width=110mm]{cnaa1a.pdf}
\end{center}
\end{frame}
\begin{frame}{Geodesics}
 \begin{itemize}
  \item For each horizontal $v$ at $P$ there are $\infty^1$ geodesics $\eta$ leaving $P$ such that $\dot{\eta}(0)=v$.  
  \item For each $\epsilon>0$ there is a geodesic leaving $O$ which is length minimizing for a time $<\epsilon$.
  \item If $\eta(0)=P$, $\dot{\eta}(0)=v$ and $\eta$ projects to a circle having radius $r>0$ we write $\eta=\eta_{P,v,1/r}$. 
 \end{itemize}
Metric $\rightarrow$ Hausdorff measures $\HHH^a$ ($a>0$) and dimensions $\text{dim}_\HHH$.
\begin{itemize}
 \item $\text{dim}_\HHH(\HH)=4$ and $d\HH^4=dxdydt$ is the Haar measure of $\HH$.
 \item $\text{dim}_\HHH(t-\text{axis})=2$ and $d\HH^2_{t-\text{axis}}=dt$ is the Haar measure of the $t$-axis.
 \item $\text{dim}_\HHH(x-\text{axis})=1$ and $d\HH^1_{x-\text{axis}}=dx$ is the Haar measure of the $x$-axis.
 \item $\text{dim}_\HHH(x,t-\text{plane})=3$ and $d\HH^3_{x,t-\text{plane}}=dxdt$ is the Haar measure of the $x,t$-plane.
 \item $\text{dim}_\HHH(x,y-\text{plane})=3$ and $d\HH^3_{x,y-\text{plane}}=\sqrt{x^2+y^2}dxdy$.
 \item Explanation: $\lambda\cdot(x,y,t)=(\lambda x,\lambda y,\lambda^2 t)$ defines the right dilations (length-areas, stratification...).
\end{itemize}
\end{frame}
\begin{frame}{Surfaces in $\HH$}
$S$ a smooth orientable surface in $\HH$; $H_P=\text{span}\{X_P,Y_P\}$; $T_PS$: the plane tangent to $S$ at $P$. 
\begin{itemize}
  \item $C\in S$ is \it characteristic \rm iff $H_C=T_CS$.
  \item Characteristic points form a small set ($\text{dim}_\HHH(\text{Characteristic set}(S))\le2$).
  \item Simply connected compact $S$'s have characteristic points.
  \item If $P\notin\text{Characteristic set}(S)$ then $\text{dim}(H_P\cap T_PS)=1$, hence
  \item $S\setminus \text{Characteristic set}(S)$ is foliated by horizontal curves.
  \item $H_P\ominus\left(H_P\cap T_PS\right)$ is the \it direction normal \rm to $S$ at $P$.
  \item If $<\cdot,\cdot>$ makes $X,Y$ into a orthonormal system for $H$, $H_P\ominus\left(H_P\cap T_PS\right)=\text{span}\{\nu_P\}$ with 
$<\nu_P,\nu_P>=1$.
  \item $\pm\nu_P$ is the \it horizontal vector normal to $S$ at its noncharacteristic point \rm $P$.
  \item Choose $\nu_P$ together with an orientation of $S$.
 \end{itemize}
\end{frame}
\begin{frame}{Surfaces in $\HH$}
\begin{center}
 \includegraphics[angle=270,width=110mm]{cnaa2.pdf}
\end{center}
\end{frame}
\begin{frame}{Distance to a surface}
 \begin{itemize}
  \item $S$: a smooth surface in $\HH$, $S=\partial\Omega$, $\Omega$ open and bounded, $\nu$ \it{in}ward horizontal normal.
  \item $d_S(P):\inf\{d(P,Q):\ Q\in S\}$.
  \item \bf Problem I: \rm smoothness properties of $d_S$?
  \item \bf Problem II: \rm given $Q$ in $S$, what can we say about the set $\NNN_QS=\{P\in\HH:\ d_S(P)=d(P,Q)\}$?
  \item $\NNN_QS$ is the \it metric normal to $S$ at $Q$. \rm
 \end{itemize}
\end{frame}
\begin{frame}{The quest for the metric normal}
\begin{center}
 \includegraphics[angle=270,width=110mm]{cnaa3b.pdf}
\end{center}
\end{frame}
\begin{frame}{Metric normal to a smooth surface}
 \begin{block}{Theorem (A., F. Ferrari)}
  $S$ a $C^1$ surface in $\HH$ and $Q\in S$. Then 
$$
\NNN_QS\subseteq\eta_{Q,\nu_QS,2/d(Q,C(\Pi_QS))}\text{ is a subarc containing }Q.
$$ 
If $S$ is $C^{1,1}$ and $Q$ in noncharacteristic, the $Q$ in the interior of the arc.

If $Q$ is characteristic, then $\NNN_QS=\{Q\}$.
 \end{block}
The \it imaginary curvature \rm of $S$ at $Q$ is $\kappa_S(Q)=2/d(Q,C(\Pi_QS))$: the curvature of the geodesic metrically normal
to $S$ at $Q$.

The \it cut-locus \rm of $S$ contains the endpoints of the geodesics' arcs $\NNN_QS$ as $Q$ varies on $S$.
\begin{center}
 \includegraphics[angle=90,width=55mm]{cnaa4.pdf}
\end{center}
\end{frame}
\begin{frame}{Metric exponential}
 $\EEE\text{xp}_S:S\times\RR\to\HH$, $(Q,\tau)\mapsto\left(\NNN_SQ\right)(\tau)=P$.
\begin{itemize}
 \item $d_S(P)=|\tau|$.
 \item \it Signed distance from \rm $S$: $\delta_S(P)=\tau:=\begin{cases}
                                  d_S(P)&\text{ if }P\in\Omega\crcr
-d_S(P)&\text{ if }P\notin\Omega
                                 \end{cases}
$
\end{itemize}
 \begin{block}{Theorem (A., F. Ferrari)}
\begin{itemize}  
\item There $\UUU$ open in $\left(S\setminus\{\text{characteristic set}\}\right)\times\RR$ such that $\EEE\text{xp}_S:\UUU\to\HH$ 
is a diffeomorphism (if $S$ is $C^{1,1}$). 
\item If $P\to\left[\EEE\text{xp}_S\right]^{-1}(P)=(Q,\tau)$, $\tau=\delta_S(P)$ and $\nabla_H\delta_S\in C(\UUU)$.
\end{itemize}
 \end{block}  
$\nabla_H f=Xf\cdot X+Yf\cdot Y$ is the horizontal gradient.
\end{frame}
\begin{frame}{The horizontal Hessian of $\delta_S$}
\begin{itemize}
\item The \it mean curvature \rm of $S$ at $Q$ is $h_S(Q)=\Delta_h\delta_S(Q)$, where $\Delta_h=XX+YY$.
 \item The horizontal Hessian of $f\HH\to\RR$ is $\text{Hess}_hf=
\begin{pmatrix}
 XXf &YXf\crcr
XYf & YYf
\end{pmatrix}.
$
\item Consider the matrices 
$I=
\begin{pmatrix}
 1 & 0\crcr
0 & 1
\end{pmatrix}
$ and 
$J=
\begin{pmatrix}
 0 & -1\crcr
1 & 0
\end{pmatrix}.
$
\end{itemize}
\begin{block}{Theorem (A., F. Ferrari)}
 $Hess_h\delta_S=\lambda_S\otimes\lambda_S\cdot\left(h_SI+\kappa_SJ\right).$
\end{block}
\begin{center}
 \includegraphics[angle=90,width=65mm]{cnaa5.pdf}
\end{center}
\end{frame}
\begin{frame}{Distance to a curve}
 \begin{itemize}
  \item $\gamma:I=(a,b)\to\HH$, $\dot{\gamma}=\alpha X+\beta Y+ mT$
  \item $d_\gamma(P)=\inf\{d(P,Q):\ Q\in\gamma(I)\}$.
  \item \bf Problem I: \rm smoothness properties of $d_\gamma$?
  \item \bf Problem II: \rm given $Q$ in $\gamma(I)$, what can we say about the set $\NNN_Q\gamma=\{P\in\HH:\ d_\gamma(P)=d(P,Q)\}$?
 \end{itemize}
\begin{block}{Straight lines.}
Suppose $\gamma$ is a straight line (i.e. a coset of a one-parameter subgroup of $\HH$).

\bf Horizontal and non-horizontal lines behave much differently. \rm 
\end{block}
\begin{itemize}
 \item $\ell_m$: straight line through $O$ with $\dot{\ell}=X+mT$.
 \item For $P_1,\ P_2$ in $\HH$: $\ell_m\cdot P_1$ and $\ell_m\cdot P_2$ are metrically parallel: $d(Q_1,\ell_m\cdot P_2)$ is independent of
$Q_1\in \ell_m\cdot P_1$.
 \item Quotient metric on $\HH/\ell_m$: $(\ell_m\cdot P_1,\ell_m\cdot P_1)\mapsto d(\ell_m\cdot P_1,\ell_m\cdot P_1)$.
\end{itemize}
\end{frame}
\begin{frame}{Projecting Heisenberg onto Gruschin}
 \begin{block}{Theorem (A., A. Baldi)}
\begin{enumerate}
 \item $(\HH/\ell_m,d)$ is isometric to the \it Gruschin plane \rm $(\RR^2,ds^2)$, $ds^2=du^2\frac{dv^2}{u^2}$.
 \item $\ell_m$ prjects to a point of the critical line $u=0$ iff it is horizontal (iff $m=0$).
\end{enumerate}
\end{block}
\begin{center}
 \includegraphics[angle=90,width=80mm]{cnaa6.pdf}
\end{center}
\end{frame}
\begin{frame}{Distance to a straight line}
 \begin{block}{Corollary}
\begin{itemize}
 \item If $\ell_m$ is not horizontal, then $d_{\ell_m}$ is smooth in a neighborhood of $\ell_m$.
\item If $\ell_m$ is horizontal, then $d_{\ell_m}$ is not smooth in any neighborhood of $\ell_m$.
\end{itemize}
 \end{block}
This leaves open the problem of understanding $\NNN_{\ell_m}Q$: the surface metrically normal to $\ell_m$ at $Q$.
\end{frame}
\begin{frame}{The quest for the surface metrically normal to a curve: non-horizontal case.}
 $\gamma:I=(a,b)\to\HH$, $\dot{\gamma}=\alpha X+\beta Y+ mT$, $m\ne0$ pointwise, $\alpha^2+\beta^2\equiv1$.  
\begin{center}
 \includegraphics[angle=90,width=80mm]{cnaa7.pdf}
\end{center}
$\eta=\eta_{Q,b}$, $b\in\TT$.
\end{frame}
\begin{frame}{Regularity of the distance function.}
 The above construction allows one to construct a \it metric exponential map \rm
\begin{block}{Exponential}
$ \EEE\text{xp}_\gamma:\gamma(I)\times\TT\times\RR^+\to\HH$, $\EEE\text{xp}_\gamma(Q,b,\tau)=\eta_{Q,b}(\tau)$.
\end{block}
\begin{theorem}
\begin{itemize}
\item The map  $\EEE\text{xp}_\gamma$ is invertible near $\gamma(I)$ and $d_\gamma(\EEE\text{xp}_\gamma(Q,b,\tau))=\tau$ for small $\tau$.
\item $d_\gamma$ is smooth ($C^1$ if $\gamma$ is $C^2$) near $\gamma(I)$.
\end{itemize}
\end{theorem}
The case of horizontal curves is quite the opposite.
\begin{theorem}
If $\gamma$ is a horizontal curve, then for all $Q$ in $\gamma(I)$ and $\epsilon>0$ there is $P$ such that $d(Q,P)<\epsilon$, but $d_\gamma$
is not differentiable (in the Euclidean case) at $P$.
\end{theorem}
\end{frame}
\begin{frame}{An application of the positive result.}
 \begin{block}{Theorem}
  Let $S=\partial\Omega$ be a compact $C^2$ surface in $\HH$ and fix $\epsilon>0$. Then there exists a $C^2$ surface $S_\epsilon=partial\Omega_\epsilon$ 
\bf without characteristic points \rm such that: 
\begin{itemize}
 \item $\HHH^4(\Omega_\epsilon\Delta\Omega)<\epsilon$;
 \item $\left|\HHH^3(S_\epsilon)-\HHH^3(S)\right|<\epsilon$.
\end{itemize}
 \end{block}
\begin{center}
 \includegraphics[angle=90,width=80mm]{cnaa8.pdf}
\end{center}
\end{frame}
\begin{frame}{An application of the negative result's proof.}
 \begin{block}{Theorem}
  Let $E\subset\HH$ be a closed subset and let $\text{Cut-locus}(E)$ be its cut-locus. Then for all open metric balls $B$ in $\HH$,
$B\cap \text{Cut-locus}(E)$ \bf is not \rm an arc of a horizontal curve.
 \end{block}
Since the cut-locus can not either have isolated points, we have the following guess.
\begin{block}{Conjecture}
For each metric open ball $B$ in $\HH$ intersecting the cut-locus of $E$ it must be $\HHH^2\left(B\cap\text{Cut-locus}(E)\right)\ge0$.
\end{block}
\end{frame}
\begin{frame}{Comments}
 \begin{itemize}
  \item Initial motivation: extending a result of Fonseca e Mantegazza from $\RR^n$ to $\HH$ (Bruno Franchi's question).
  \item Some of the above is proved for more general Carnot groups in joint work with Ferrari e Montefalcone.
  \item A more general study of the cut-locus might be interesting.
  \item Ferrari e Valdinoci have interesting applications of some of the above to some nonlinear PDE's.
  \item Most results await sharp regularity versions of themselves.
 \end{itemize}
\end{frame}

\begin{frame}
 \centerline{\large Happy birthday Gianni!}
\end{frame}




\end{document}

\begin{frame}{Motivation}
 Fonseca and Mantegazza:
\begin{itemize}
 \item $\Omega\in\RR^n$ open, bounded.
 \item $\WWW(t)=t^2(1-t)^2$.
 \item For $\epsilon>0$ and $u\in L^1(\Omega)$ let
$$
\FFF_\epsilon(u):=\begin{cases}
                  \int_\Omega\left(\frac{\WWW(u(x))}{\epsilon}+\epsilon^3\left\|\text{Hess}u(x)\right\|^2\right)dx&\text{if }u\in W^{2,2}(\Omega),\crcr
+\infty & \text{otherwise}.
                 \end{cases}
$$
 \item For $u\in L^1(\Omega)$ let
$$
\FFF(u)=:\begin{cases}
                  {\bf m}\Vert \partial \{u=0\} \Vert_{\RR^n}(\Omega) , &\mbox{if}\:\:u\in BV_{\RR^n}(\Omega ,\{0,1\}),\crcr
+\infty & \text{otherwise},
                 \end{cases}
$$
 \item where $\bf{m}:=\min\left\{\int_\RR\left(\WWW(t)+|f^{\prime\prime}(t)|^2\right)dt:\ f\in W^{2,2}(\RR),\right.$
$\left.\lim_{t\to+\infty}f(t)=1,\ \lim_{t\to-\infty}f(t)=0\right\}$.
\end{itemize}
\end{frame}

