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\begin{document}
\title{The double bubble conjecture} 
\author{Joel Hass} %\thanks{Partially supported by the NSF.}
\address{ Department of Mathematics,
University of California, Davis, CA 95616}
\email{hass@@math.ucdavis.edu} 
\author{Michael Hutchings} %\thanks{Supported by an NSF Graduate Fellowship.} 
%\email{hutching@@math.harvard.edu} 
\address{Department of Mathematics,  Harvard University, Cambridge, MA  02138}
\email{hutching@@math.harvard.edu}
\author{ Roger Schlafly }
\address{ Real Software, PO Box 1680, 
Soquel, CA 95073}
\email{rschlafly@@attmail.com} 
\thanks{Hass was partially supported by the NSF}
\thanks{Hutchings was supported by an NSF Graduate Fellowship.}
\date{September 11, 1995}
\subjclass{Primary 53A10; 49Q10, 49Q25}
\keywords{Double bubble; isoperimetric}
\communicated_by{Richard Schoen}
\setcounter{page}{98}

\renewcommand{\currentvolume}{1}
\renewcommand{\currentissue}{3}

\setcopyrightline{Hass, Hutchings, Schlafly}

%\maketitle
\begin{abstract}
The classical isoperimetric inequality states that the surface of
smallest area enclosing a given volume in $R^3$ is a sphere. We show that the
least area
surface enclosing two equal volumes is a double bubble, a surface made of two
pieces of
round spheres separated by a flat disk, meeting along a single circle at an
angle of 
$2 \pi / 3$.
\end{abstract}
\maketitle 
\newtheorem{thm}{Theorem}
\newtheorem{conj}{Conjecture}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}

\section{Introduction}

\vskip.15in 

\begin{verse} 
\em \hskip2.4in Double, double, toil and trouble,\\
\hskip2.4in Fire burn and cauldron bubble. 
\end{verse}  
\hskip3in Macbeth Act 4, Scene 1, Line 10


\vskip.2in 

The double bubble is the surface in $R^3$ obtained by taking two pieces of
round
spheres separated by a flat disk, meeting along a single circle at an angle of
$2 \pi /
3$.  It has long been thought that the double bubble minimizes area
among all piecewise-smooth surfaces enclosing two equal volumes.  

Experimental evidence towards this conjecture can be obtained by blowing soap
bubbles and observing the resulting shapes.  If one blows two soap bubbles of
equal
size and pushes them together until they conglomerate to form a compound bubble,
one
obtains a double bubble.  Such experiments were carried out by the Belgian
physicist J.
Plateau in the middle of the 19th century.    Plateau established experimentally
that a soap
bubble cluster is a piecewise-smooth surface having only two types of
singularities. The
first type of singularity occurs when three smooth surfaces come together along
a smooth
triple curve at an angle of $120^o$.  The second type of singularity occurs when
six smooth
surfaces and four triple curves converge at a point, with all angles equal.  The
angles are
equal to those of the cone over the 1-skeleton of a regular tetrahedron. 

C.V. Boys, discussing the work of Plateau in his famous book on soap bubbles
\cite{Boys}
writes,

``When however the bubble is not single, say two have been blown in real contact
with one
another, again the bubbles must together take such a form that the total surface
of the two
spherical segments and of the part common to both, which I shall call the
interface, is the
smallest possible surface which will contain the two volumes of air and keep
them separate."

We have obtained a proof of this conjecture for the case of two equal volumes. 

\begin{thm} \cite{H-S} The double bubble uniquely minimizes area
among all surfaces in $R^3$ enclosing two equal volumes.
\end{thm}

We remark that a planar analogue has recently been solved in \cite{{ABFHZ}}.

Our result can also be viewed as an isoperimetric inequality.

\begin{cor}
For any surface in $R^3$ enclosing two regions, each having volume $V$,
the area $A$ satisfies 
$$
A^3 \ge 243 \pi V^2
$$
with equality if and only if it is isomorphic to the standard symmetric
double bubble enclosing two regions of volume $V$ by an isometry of $R^3$.
\end{cor}

This result gives the first explicit example of closed minimizing surfaces in
$R^3$ which
exhibits any of the singularities predicted by Plateau.

\vskip3em

\section{An outline of the proof }

Existence and regularity of a minimizer were established by F. Almgren and J.
Taylor. 
Almgren showed in \cite{A} that there exists an area minimizing surface in $R^3$
among
the set of surfaces enclosing a given pair of volumes.  Here {\em surface}
refers to a generalized notion used in geometric measure theory, which includes
piecewise-smooth surfaces.  Almgren showed that the solution is a smooth
surface
almost everywhere.  Taylor obtained additional information on the nature of the
singularities \cite{T}.  She showed that a minimizer is a piecewise-smooth
surface whose singularities consist of smooth triple curves along which
three smooth surfaces come together at an angle of $120^o$, and isolated points
where pieces
of surface converge.  At these isolated points the asymptotic cone is the cone
over the
1-skeleton of a regular tetrahedron.  

Our proof that the double bubble minimizes is by a direct computational
attack on the space of surfaces.  The space of surfaces enclosing two equal
volumes is
infinite dimensional. By a series of analytic and geometric arguments this space
is
reduced first to a union of finite dimensional sets, then a compact
two-dimensional set, and
ultimately the conjecture is reduced to a finite number of numeric
computations.

It is a classical result that any surface minimizing area while enclosing a
given volume has
constant mean curvature on each smooth piece.  The second ingredient in our
proof is a
general theorem about symmetry in soap bubble clusters. 

\begin{thm} 
An area minimizing enclosure of $m$ volumes in $R^n$, for $ m<n$, is
rotationally symmetric
about an $(m-1)$-dimensional plane.
\end{thm}

The ideas behind this theorem are due to Brian White and Frank Morgan, and
versions of it are
written in \cite{Fo}, \cite{Morgan} and \cite{Hu}.    It implies the classical
isoperimetric
theorem, and also tells us that a minimizing bubble enclosing two given regions
in $R^3$ is a
surface of revolution.  Constant mean curvature surfaces of revolution in $R^3$
were
classified by Delaunay \cite{D}, and they form the pieces of our minimizing
bubble.  

Almgren's theorem provides no information about the topological
complexity of the possible solutions.  The existence theorem allows for the
strange
possibility that the volumes enclosed may be disconnected.  Even the exterior
region may
be disconnected, in which case there are ``empty regions'' which do not
contribute to
either of the two volumes we are enclosing.  The main tool in controlling the
topology is:

\begin{thm} \cite{Hu} \label{concavity}
If $A(V_1,V_2)$ is the minimum area for surfaces in $R^3$ enclosing
volumes $V_1$ and $V_2 $, then $A$ is concave as a function of $V_1$ and $V_2
$.
\end{thm}

The basic ingredient in the proofs of the symmetry theorem and the concavity
theorem is the idea of ``symmetrization''. Suppose we have a soap bubble
cluster
and a hyperplane. This hyperplane divides the cluster into two halves. We can
replace one half with the reflection of the other half across the hyperplane.
The area (respectively volume) of the original cluster is the average of the
area
(respectively volume) of the two different symmetrizations.

In the special case when the hyperplane bisects both enclosed volumes and the
cluster is area minimizing, both symmetrizations are minimizers, as otherwise
one
would have too little area. It follows that the cluster is orthogonal to the
hyperplane, since otherwise the symmetrizations would have corners which could
be smoothed to decrease area. Now the Borsuk-Ulam Theorem provides hyperplanes
bisecting the volumes of a cluster, and one can find the axis of symmetry as an
intersection of such hyperplanes.

If we could find hyperplanes dividing the volumes of a minimal cluster into
other proportions, this would immediately imply concavity, since the areas of
the symmetrizations must be greater than or equal to the areas of the
minimizers
for those volumes. The Borsuk-Ulam Theorem only allows us to bisect the
volumes. However, another topological argument shows that if concavity fails,
then there are extra hyperplanes of symmetry. In particular we find that a
minimal bubble enclosing two volumes would have to be a union of concentric
spheres, which is
clearly not area minimizing.

To illustrate how concavity applies to connectedness, we can quickly
deduce that there are no empty chambers.  Concavity, together with the fact
that $A(V_1,V_2) \to\infty$ as $V_1 \to\infty$ (by the isoperimetric theorem,
since an
enclosure of volumes $V_1$ and $V_2$ is also an enclosure of volume $V_1$),
implies that
$A(V_1,V_2)$ is a strictly increasing function of $V_1$ for $V_2$ fixed.
Now, if an area-minimizing enclosure of volumes $V_1$ and $V_2$ has an empty
chamber
of volume $E$, then this is also an enclosure of volumes $V_1+E$ and $V_2$.
Then $A(V_1+E,V_2)\le A(V_1,V_2)$, a contradiction.

From Theorem~\ref{concavity} we deduce that the volumes are connected, and hence
the
minimizer has either the topology of the double bubble or of one other possible
configuration.  A {\em torus bubble} is a surface of revolution constructed by
taking two
circular arcs of the same radius, facing each other, each with one endpoint and
center on the
$x$-axis, and connecting the other endpoints with Delaunay curves meeting at 120
degrees.  We
then get a bubble surrounding two components, one homeomorphic to a torus and
one
homeomorphic to a ball.  It is possible, though not immediately clear, to make
such a
construction so that the curves meet at $120^o$ angles, so that torus bubbles
do indeed exist.

The possible torus bubbles may be parameterized as follows:  choose a
radius $ r > 0$ for the arcs, angles $\theta_1$ and $\theta_2$ subtending
the arcs, and mean curvature $H_i$ for the inner Delaunay surface.  Then
the spherical pieces have mean curvature $2/r$ and the outer Delaunay
surface must have mean curvature $H_o = 2/r - H_i \ge 0$.  The Delaunay
curves are then determined by an ordinary differential equation
and the initial conditions at either endpoint.

Geometric arguments show that given $r, \ \theta_1$, and $H_i$,
there are at most two values of $\theta_2$ for which the curves can meet at the required $120^o$ angles, and these can be
obtained algebraically by solving a quadratic equation.  Perturbation arguments
restrict the
values for $\theta_2$ that can occur in a minimizing bubble.   One of these
values is equal to
$\theta_1$ which gives rise to a symmetric torus bubble.  We show that such
symmetric
bubbles are always unstable.  Thus, the torus bubble is determined
by $r, \ \theta_1$, and $H_i$, and we can assume by scaling and reflection
that $r = 1$ and $\theta_1  <  \theta_2$.

We next do a computation to show that torus bubbles cannot be minimizers.
The idea is to make an exhaustive search of all possible
$\theta_1, H_i,$
where $0 \le \theta_1 \le \pi$ and $H_i \le 2$.  In each case, we show that either
$\theta_2$
does not exist in the appropriate range, or that the two Delaunay surfaces
forming the
boundary of the torus region do not match up when integrated, or that the two
regions in the
torus bubble have unequal volumes.  It turns out that there are one-parameter
families of
torus bubbles which are critical points of the area function, but 
that if they enclose equal volumes then there is always a
perturbation that will decrease their area while preserving both volumes.

The computation involves thousands of numerical integrations to get precise
information
about Delaunay surfaces.  We use IEEE double precision arithmetic and interval
arithmetic to
derive strict bounds for all the estimates and calculations
\cite{ANSI},\cite{Moore}.  A
detailed proof appears in \cite{H-S}.

\medskip
{\em We are indebted to Frank Morgan for introducing us to this problem and to
Morgan and W. Kahan for helpful discussions.}

%\vskip1.5cm

\begin{thebibliography}{HHH}
%\vskip.5cm

\bibitem{ABFHZ} M. Alfaro, J. Brock, J. Foisy, N. Hodges and J. Zimba {\em The
standard
double soap bubble in $ R^2$ uniquely minimizes perimeter,} Pac. J. Math. 159,
47-59 (1993). \MR{94b:53019}
\bibitem{A} F.J. Almgren, {\em Existence and regularity almost everywhere of
solutions to
elliptic variational problems with constraints,} Memoirs Amer. Math. Soc. 4,
165-199
(1976). \MR{54:8420}
\bibitem{AT} F.J. Almgren and J. Taylor, {\em The geometry of soap films and
soap bubbles,}
Sci. Amer. 235,82-93 (1976). 
\bibitem{ANSI} ANSI/IEEE {\em Standard 754-1985 for Binary Floating-Point
Arithmetic,}
The Institute of Electrical and Electronic Engineers, New York, 1985.
\bibitem{Boys} C.V. Boys, {\em Soap Bubbles,} Dover Publ. Inc. NY 1959 (first
edition
1911).
\bibitem{D} C. Delaunay, {\em Sur la surface de revolution dont la courbure
moyenne est
constante,} J. Math. Pure et App. 16, 309-321 (1841).
\bibitem{Eells} J. Eells, {\em The surfaces of Delaunay,} Math. Intelligencer 9,
53-57 (1987). \MR{88h:53011} 
\bibitem{Fo} J. Foisy, {\em Soap bubble clusters in $R^2$ and $R^3$,}
undergraduate
thesis, Williams College (1991).
\bibitem{H-S} J. Hass and R. Schlafly, {\em Double Bubbles Minimize,}
(preprint).
\bibitem{Hu} M. Hutchings, {\em The structure of area-minimizing double
bubbles,}
to appear in J. Geom. Anal.
\bibitem{Morgan} F. Morgan, {\em Clusters minimizing area plus length of
singular curves,}
Math. Ann. 299, 697-714 (1994). \MR{95g:49083}
\bibitem{Moore} R. E. Moore, {\em Methods and Applications of Interval
Analysis,}
SIAM, 1979. \MR{81b:65040}
\bibitem{Pl} J. Plateau, {\em Statique exp\'erimentale et th\'eorique 
des liquides
soumis
aux seules forces moleculaires,}  Gathier-Villars, Paris, 1873.
\bibitem{T} J. Taylor, {\em The structure of singularities in soap-bubble-like
and
soap-film-like minimal surfaces,} Ann. of Math. 103, 489-539 (1976). 
\MR{55:1208a}\newpage
\bibitem{Th} D'arcy Thompson, {\em On growth and form,} Cambridge Univ. Press,
NY, 1959,
(first edition 1917). \MR{23:B1601}

\end{thebibliography}

\end{document}
