\documentstyle[12pt,graphics,amsfonts,epsf]{article}
% version history: 
%		
%	
\textheight=574pt
\textwidth=432pt
\oddsidemargin=18.88pt
\evensidemargin=18.88pt
\topmargin=14.21pt

\newcommand{\R}{\bf R}
\newcommand{\C}{\bf C}
\newcommand{\cliff}{{\cal C}}


\newtheorem{lemma}{Lemma}
\newtheorem{exercise}[lemma]{Exercise}
\newtheorem{claim}[lemma]{Claim}
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}[lemma]{Proposition}
\newtheorem{definition}[lemma]{Definition}


\begin{document}

\nocite{*}

\title{Positively curved surfaces \\ with no tangent support plane}
\author{John McCuan\thanks{The author thanks the National Science Foundation 
for the postdoctoral fellowship which supported this work and the University of California, Berkeley and the Mathematical Sciences Research Institute for 
their hospitality.}}

%\address{John McCuan\\Mathematics Department\\University of California, Berkeley\\Berkeley, CA 94720}
\makeatletter
%\email{johnm@math.berkeley.edu}
\makeatother

\maketitle

\begin{abstract}
We discuss a characterization of positively curved surfaces $M$ with the 
property that at each point the tangent plane to $M$ is not a 
support plane for the entire surface.  
A one parameter family of examples which have special relevance 
with respect to the 
characterization is also given.  Each member of this 
family is a smooth embedded surface in ${\Bbb R}^3$ that is 
topologically a disk, has everywhere positive Gauss curvature, but has 
none of its tangent planes as a support plane.
\end{abstract}

\pagestyle{headings}
{\makeatletter
\gdef\@evenhead{\thepage\hfil\small\sc John McCuan}%
\gdef\@oddhead{%
\ifodd\thepage
\hfil{\small\uppercase{Positively curved surfaces with no tangent support plane}}\hfil
\thepage
\else{\thepage\hfil\small\sc \uppercase{John McCuan}\hfil}\fi}
}

%\maketitle

\section*{Introduction}
Throughout this paper $M$ denotes a smooth positively curved immersion into 
${\Bbb R}^3$ of a compact 
connected surface which may have a (smooth) boundary.  
%We denote the convex hull of $\partial M$ by $K$.

The condition of positive curvature implies that $M$ is {\em strictly 
locally convex,\/} i.e., the plane $T_pM$ tangent to $M$ at $p$ is a 
support plane for some (intrinsic) neighborhood $N$ of $p$, and 
$N \cap T_pM  = \{p\}$.  A well known theorem of Hadamard \cite{HadSur} 
asserts that if % $M$ is compact and 
$\partial M = \phi$, then 
$M$ is (globally) the boundary 
of a (strictly) convex region in ${\Bbb R}^3$.  Various 
authors have generalized Hadamard's theorem.  
%In particular, 
%Chern and Lashof \cite{CheTot} relax the condition of positive Gauss 
%curvature to nonnegative Gauss curvature, do Carmo and Lima \cite{doCImm} 
%admit complete, non-compact surfaces, and van Heijenoort \cite{HeiLoc} 
%considers non-smooth locally convex surfaces.  
We mention in particular that do Carmo and Lima \cite{doCImm} draw 
essentially the 
same conclusion for complete, non-compact surfaces, though in this case 
the region in ${\Bbb R}^3$ is non-compact.  

Recently % in his thesis 
Mohammad Ghomi extended Hadamard's theorem to the case when 
$\partial M \ne \phi$.  When does such a surface lie in the boundary 
of a strictly convex region?  One sees immediately the following necessary 
condition.  For each $p\in\partial M$ the tangent plane $T_pM$ 
is a support plane for $\partial M$, and 
$\partial M \cap T_pM = \{ p\}$.  We 
say that $\partial M$ is {\em convex compatible\/} if this condition holds 
component-wise, i.e., for each component 
$C$ of $\partial M$ and each $p\in C$ we have that $T_pM$ is a support plane 
for $C$ and $C \cap T_pM = \{ p\}$.  Ghomi proves this latter condition is 
sufficient for $M$ to lie in the boundary of a convex region.  
\begin{theorem}[Ghomi \cite{GhoDis}]  \label{ghomi} 
A compact, positively curved surface $M$ has convex compatible 
boundary if and only if 
% $\partial M$ is convex compatible, then 
$M$ is a subset of the boundary of a 
convex region in ${\Bbb R}^3$.  
\end{theorem}
In particular, every tangent plane to $M$ is a support plane for $M$.
%Ghomi points out that $T_pM$ need not be a support plane for $M$ in 
%general, see Figure~\ref{ghomi.ex}.  He conjectures however that at 
%least one tangent plane to the surface will be a support plane.  We give a 
%counterexample which is an embedded topological disk.

\medskip

%More generally, we call $M$ an {\em everted positively curved surface\/} 
%(e.p.s.) if none of the tangent planes $T_pM$ is a support plane for $M$.  
In the case $\partial M\ne \phi$ we denote the convex hull of 
$\partial M$ by $K$.  Following Ghomi, we say $\partial M$ is {\em convex} 
if $\partial M \subset \partial K$.  This condition clearly holds for many 
positively curved surfaces.  In particular, if $M$ ``bulges out'' so that 
each interior point of $M$ is in ${\Bbb R}^3\backslash K$, then 
$\partial M$ is convex, n.b., Figure~\ref{fig1}.  
\begin{figure}[hb]
\centerline{
\begin{picture}(140,140)
	\put(0,0){\epsfxsize=2in\epsffile{figures/convbd.ps}}
\end{picture}}
\centerline{ }
\caption{Positively curved surface with boundary.}
\label{fig1}
\end{figure}
It was observed in \cite{GhoDis} that this condition 
% the convexity of $\partial M$ 
% does not imply the conclusion of 
%Theorem~\ref{ghomi}. 
is too weak to imply $M$ lies in the boundary of a convex region.  
(This also is illustrated in Figure~\ref{fig1}.)  
Convexity of the boundary is enough, however, 
to imply the existence of 
at least one tangent support plane.  This follows from a recent 
characterization of positively curved surfaces with convex boundary.  
\begin{theorem}[Alexander-Ghomi \cite{GhoSay}]  
$\partial M$ is convex if and only if 
\begin{equation}\label{1}
{\rm int\,}M \subset {\Bbb R}^3\backslash K.
\end{equation}
\end{theorem}
Thus, if $\partial M$ is convex, then for some $p\in {\rm int\,} M$, 
we have $d(p,K) = \max_q d(q,K)$.  It follows that 
$T_pM$ is a support plane for $M$.

\pagebreak

The same reasoning shows that {\em if $M$ has no tangent support plane,} 
then 
\begin{equation}\label{2}
{\rm int\,} M \subset {\rm int\,} K.
\end{equation}
Under the same assumption on $M$,
\begin{equation}\label{3}
T_pM \cap {\rm int\,} K \ne \phi \ {\rm for\  all\ } 
p\in \partial M.
\end{equation}
Otherwise, $T_pM$ is a support plane for $K$ and hence for $M$ by (\ref{2}).

Since (\ref{1}) is known to hold for a reasonably large class of surfaces 
and (\ref{2}) is quite the opposite property, one might guess that at 
least one of (\ref{2}) and (\ref{3}) is always violated.  Presumably, such 
considerations led to the conjecture in \cite{GhoDis} that every positively 
curved immersion $M$ has at least one tangent support plane.  We describe 
below a counterexample which is an embedded topological disk.

Notice that (\ref{2}) and (\ref{3}) characterize the positively curved 
surfaces with no tangent support plane.  More generally, 
these surfaces are characterized 
by the condition that for some $A\subset {\rm int\,} M$,
\begin{equation}\label{altern}
A \subset {\rm int\,} K\quad {\rm and}\quad T_pM \cap {\rm int\,} K \ne 
\phi \ {\rm for\ all}\ p\in M\backslash A.
\end{equation}
From this observation we 
obtain the following lemma. % which simplifies the discussion below.
\begin{lemma} \label{lem1}
If $M$ is a positively curved immersion with no tangent support plane, 
and $\tilde{M} \subset M$ is a surface with boundary 
such that the convex hull $\tilde{K}$ of $\partial\tilde{M}$ is $K$, then 
$\tilde{M}$ has no tangent support plane.  In particular, if $B\subset 
\partial M$ has convex hull $K$ and $B\subset \tilde{M}$, then the 
conclusion holds.
\end{lemma}
Aside from simplifying the discussion below, this lemma shows the 
importance of the convex hull $K$.  In our example, the set $B$ that 
generates $K$ consists of ten small arcs which, with obvious modifications, 
may be made arbitrarily small.

It should also be remarked that there are simpler examples than the 
one described below.  (David Hoffman and Jim Hoffman have suggested 
two such constructions.  See the schematic diagrams at the 
end of the paper.)  We felt, however, that our example illustrated 
dramatically the significance of the convex hull $K$ of $\partial M$.

\medskip

Now we show that our remarks are not vacuous.

\section*{Examples}
The starting point for our construction is the following lemma which is 
verified by an elementary calculation.
\begin{lemma}\label{lem2}
Let $\gamma=\gamma(s)$ be a smooth planar curve in 
$\{(x,y,0)\}\subset {\Bbb R}^3$ 
with nonvanishing curvature $\kappa = \kappa(s) = |\ddot{\gamma}|$ and 
unit normal $n=\ddot{\gamma}/|\ddot{\gamma}|$.  

Let $T^+(s)$ and $T^-(s)$ define a domain 
$\Sigma = \{ (s,t): T^-(s) < t < T^+(s)\}$.  

Let $M^+ = \max_sT^+$ and $M^- = \min_sT^-$, and assume that 
$f:[M^-,M^+] \to [0,\infty)$ is a smooth function satisfying $f(0) = 0$ 
and $f'' > 0$.  Assume also that 
$f\circ T^+(s),f\circ T^-(s) < 1/\kappa(s)$ for all $s$.  

Then $X(s,t) = \gamma(s) + fn + te_3$ gives a well defined positively 
curved immersion $M$ of $\Sigma$ into ${\Bbb R}^3$ where 
$e_3 = (0,0,1)$.
\end{lemma}

We apply Lemma~\ref{lem2} with $\gamma$ given by the ``spirograph curve'' 
\[
	\gamma(\theta) = (3\cos\theta + \rho \cos(3\theta/2),
		3\sin\theta - \rho \sin(3\theta/2))
\]
determined by a point rigidly fixed at a distance $\rho$ from the center 
of a circle of radius $2$ which is rolling inside a circle of radius $5$; 
see Figure~\ref{figspiro}(a).  
We assume initially that $2 < \rho \le 5/2$.  
An elementary calculation then shows 
\[
\kappa(\theta) = {3\rho^2 - 8 - 2\rho\cos(5\theta/2) 
		\over{3(\rho^2 + 4 - 4\rho\cos(5\theta/2))^{3/2}}} >0.
\]
% We also record for later reference that 
%\[
%	|\gamma| = (\rho^2 + 9 + 6\rho\cos(5\theta/2))^{1/2}.
%\]

Let $T^\pm(\theta) = \pm [A + \epsilon - (A - \epsilon)
		\cos(5\theta/2)]$ where $0<\epsilon \le 1/2\sqrt{\kappa(0)} 
< A \le 1/2\sqrt{\kappa(2\pi/5)}$.  This choice 
defines a strip $\Sigma$ as in Figure~\ref{strip}(a).  
\begin{figure}[hb]
\centerline{
\begin{picture}(340,100)
	\put(0,20){\epsfxsize=2in\epsffile{figures/strip2.ps}}
	\put(65,0){(a)}
	\put(200,20){\epsfxsize=2in\epsffile{figures/cutstrip3.ps}}
	\put(265,0){(b)}
\end{picture}}
\centerline{ }
\caption{Strip between two periodic curves.}
\label{strip}
\end{figure}

Finally, we take $f(t) = t^2$, and taking account of periodicity in 
Lemma~\ref{lem1}, we obtain an annular immersion parameterized by 
\[
	X(\theta,t) = \gamma + t^2 n + te_3.
\]

The resulting surface $M$ is composed of five (positively curved) taco 
shells (see Figure~\ref{figimmers}).  Unlike standard flat taco shells, 
these curl at the corners.  Because of this, they may be smoothly 
concatenated so that their backs lie successively along the spirograph 
curve, and their wings are directed outward.  The projection into the 
$x,y$-plane of $\partial M$ is shown in Figure~\ref{figspiro}(b), 
\begin{figure}[ht]
\centerline{
\begin{picture}(305,140)
	\put(0,10){\epsfxsize=1.5in\epsffile{figures/spiro.ps}}
	\put(200,10){\epsfxsize=1.5in\epsffile{figures/boundary-proj.ps}}
	\put(41,-10){(a)}
	\put(241,-10){(b)}
\end{picture}}
\centerline{ }
\caption{Spirograph curve and the projection of $\partial M$.}
\label{figspiro}
\end{figure}
and a closeup of the corners where two taco shells join is shown in 
Figure~\ref{figcorner}(a)
\begin{figure}[hbt]
\centerline{
\begin{picture}(300,150)
	\put(0,-4){\epsfxsize=1.2in\epsffile{figures/taco-corner.ps}}
	\put(90,-19){\epsfxsize=1in\epsffile{figures/corn2.ps}}
	\put(70,-7){(a)}
	\put(220,-4){\epsfxsize=1.2in\epsffile{figures/corner-cutaway.ps}}
	\put(240,-7){(b)}
\end{picture}}
\centerline{ }
\caption{(a) Where two tacos meet ($-\pi/8 \le \theta \le \pi/8$). (b) The 
desingularization.}
\label{figcorner}
\end{figure}

The convex hull $K$ of $\partial M$ is generated by ten arcs on the 
extreme edges of the wings.  These arcs are quite small when $A$ is 
large (say $\ge 2$), though Figure~\ref{figimmers} corresponds to 
a considerably lower value of $A$.  The fact that the arcs are small 
can be used to give a proof that $M$ has no tangent support 
plane as follows.

The largest $z$-values (in magnitude) are obtained when 
$\kappa$ is at its minima and $t=T^{\pm}(\theta)$.  These boundary 
values are found to be at $\theta_k = 2\pi/5 + 4\pi k/5$ for 
$k\in {\Bbb Z}$ \pagebreak
\begin{figure}[htb]
\centerline{
\begin{picture}(340,390)
	\put(0,320){\epsfxsize=2.5in\epsffile{figures/taco.ps}}
	\put(200,320){\epsfxsize=2.5in\epsffile{figures/twotaco.ps}}
	\put(0,160){\epsfxsize=2.5in\epsffile{figures/threetaco.ps}}
	\put(200,160){\epsfxsize=2.5in\epsffile{figures/fourtaco.ps}}
	\put(0,0){\epsfxsize=2.5in\epsffile{figures/fivetaco.ps}}
	\put(220,0){\epsfxsize=2in\epsffile{figures/cleanview.ps}}
\end{picture}}
\centerline{ }
\caption{Construction of the immersed annular example.}
\label{figimmers}
\end{figure}
and are given by the ten points
\[
	X(\theta_k,T^{\pm}(\theta_k)) = \gamma(\theta_k) + {T^+(\theta_k)}^2n 
		\pm  T^+(\theta_k) e_3, \qquad 
		k = 0,\pm1,\pm2.
\]
The convex hull $K_0$ of $\{X(\theta_k,T^{\pm}(\theta_k))\}$ is a 
pentagonal prism, and 
$K_0 \subset K$.  One can verify directly that, for large enough 
values of $A$, most interior 
points of $M$ are in 
${\rm int\,} K_0$.  More precisely, if $A\ge 2$, then 
the condition $|\theta - \theta_k| >\pi/8$ for $k = 0, \pm 1, \pm 2$ implies 
\begin{eqnarray}
	|\gamma(\theta) + t^2n| & \le & |\gamma(\theta)| + T^+(\theta)^2 
		\nonumber\\
		& < & r_0 \equiv (3-\rho + 4A^2) \cos(\pi/5). \label{star}
\end{eqnarray}
The number $r_0$ in this inequality is the radius of the circular cylinder 
inscribed in $K_0$.  Thus, $X(\theta,t) \in {\rm int\,} K_0$.

\thispagestyle{empty}

%\pagebreak

%We claim first that for some $\delta > 0$ the 
%set $A = \{X(\theta,t): -\pi/5+2\pi k/5-\delta < \theta < \pi/5 + 2\pi k/5 
%		-\delta,\  k=0,\pm1,\pm2 \}$ is a subset of ${\rm int\,}K_0$.  
%Secondly, we claim that ${\rm int\,}M \backslash A \subset {\rm int\,}K$.  
%Finally, we claim that $T_pM \cap {\rm int\,}K_0 \ne \phi$ for 
%$p\in \partial M\backslash A$.  These claims establish that $M$ is an 
%e.p.s. by Lemma~\ref{lem1}.

%\pagebreak

For the remaining points $p = X(\theta,t) = \gamma + 
t^2 n + t e_3$ with $|\theta - \theta_k| \le \pi/8$ 
for some $k = 0,\pm1,\pm2$, we show that 
$T_pM \cap {\rm int\,} K_0 \ne \phi$.  The line 
through $p$ with tangent direction $X_t = 2tn + e_3$ 
comes closest to the origin at the point 
\[ 
	q = \gamma -t^2\left( {1+4\gamma\cdot n
		\over{4t^2+1}}\right) n + 2t \left( 
		{t^2 -\gamma\cdot n\over{4t^2 +1}}
		\right) e_3
	\in T_pM.
\] 
If we specialize to $\rho = 5/2$, then it follows 
that $0\le \gamma\cdot n \le 1/2$ and, hence, that 
the third component of $q$ is in the open interval 
$(T^-(\theta), T^+(\theta))$.  Consequently, we only 
need to check that the projection 
\[
	q_0 = \gamma -t^2\left( {1+4\gamma\cdot n
		\over{4t^2+1}}\right) n
\]
lies in the pentagonal base of $K_0$.  It is easily 
checked under our present assumptions, $\rho = 5/2$, 
$A\ge 2$, and $|\theta - \theta_k| \le \pi/8$, 
that $|\gamma| \le 3$.  From this it is easy to 
see that $|q_0| < r_0$ where $r_0$ is given in 
(\ref{star}).  Thus, $q \in T_pM\cap {\rm int\,}K_0$.

By the characterization (\ref{altern}), we conclude 
that $M$ has no tangent support plane.

By removing portions of $\Sigma$ as indicated in 
Figure~\ref{strip}(b), we obtain an embedded disk which, since the 
dark curves correspond to a set $B$ whose convex 
hull is $K$, has no tangent support plane 
by Lemma~\ref{lem1}.  Two views of this surface are shown in 
Figure~\ref{disk}.
\begin{figure}[h]
\centerline{
\begin{picture}(350,170)
	\put(0,0){\epsfxsize=2.5in\epsffile{figures/disk2.ps}}
	\put(220,0){\epsfxsize=2in\epsffile{figures/disktop2.ps}}
\end{picture}}
\centerline{ }
\caption{An embedded topological disk.}
\label{disk}
\end{figure}

\pagebreak

David Hoffman and Jim Hoffman \cite{HofSay} have given other examples 
(Figure~\ref{hof}).  David's example consists of two portions of 
spheres connected by a strip of positive curvature like that described 
in Lemma~\ref{lem2}.  Jim's is a positively curved ribbon that 
nearly lies on a catenoid.  
\begin{figure}[hb]
\centerline{
\begin{picture}(350,80)
	\put(0,-10){\epsfxsize=2in\epsffile{figures/hoff2.ps}}
	\put(50,-10){David}
	\put(250,-10){Jim}
	\put(200,-5){\epsfxsize=1in\rotatebox{90}{\epsffile{figures/ribbon.ps}
}}
\end{picture}}
\centerline{ }
\caption{Other examples.}
\label{hof}
\end{figure}



% One can also use the characterization (\ref{altern}) to show that this 
% surface has no tangent support plane and it can be made topologically 
% trivial as well.  
% We felt, however, that our original example illustrated 
% dramatically the significance of the convex hull $K$ of $\partial M$.


\bibliographystyle{plain}
\bibliography{bib/ovaloid}

\begin{thebibliography}{1}

\bibitem{CheTot}
S.S. Chern and R.K. Lashof.
\newblock On the total curvature of immersed manifolds.
\newblock {\em Michigan Math. J.}, 5:5--12, 1958.

\bibitem{doCImm}
M.~do~Carmo and E.~Lima.
\newblock Immersions of manifolds with non-negative sectional curvatures.
\newblock {\em Bol. Soc. Brasil Mat.}, 2:9--22, 1971.

\bibitem{GhoSay}
M.~Ghomi.
\newblock Personal communication on work with S. Alexander.

\bibitem{GhoDis}
M.~Ghomi.
\newblock {\em Strictly Convex Submanifolds And Hypersurfaces Of Positive
  Curvature}.
\newblock PhD thesis, Johns Hopkins, 1998.

\bibitem{HadSur}
J.~Hadamard.
\newblock Sur certaines propriet\'{e}s des trajectoires en dynamique.
\newblock {\em J. Math. Pures Appl.}, 3:331--387, 1897.

\bibitem{HofSay}
D.~Hoffman and J.~Hoffman.
\newblock Personal communication.

\bibitem{HeiLoc}
J.~van Heijenoort.
\newblock On locally convex manifolds.
\newblock {\em Comm. Pure Appl. Math.}, 5:223--242, 1952.

\end{thebibliography}

\hskip-\parindent
John McCuan\\Mathematics Department\\University of California,
Berkeley\\Berkeley, CA 94720\\johnm@math.berkeley.edu 
\end{document}


