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\title{The double bubble conjecture} 
\author{Joel Hass} %\thanks{Partially supported by the NSF.}
\address{ Department of Mathematics,
University of California, Davis, CA 95616}
\author{Michael Hutchings} %\thanks{Supported by an NSF Graduate Fellowship.} 
\address{Department of Mathematics,  Harvard University, Cambridge, MA  02138}
\author{ Roger Schlafly }
\address{ Real Software, PO Box 1680, 
Soquel, CA 95073}
\thanks{Hass was partially supported by the NSF}
\thanks{Hutchings was supported by an NSF Graduate Fellowship.}
\date{September 11, 1995}
\subjclass{53A10; 49Q10, 49Q25}
\keywords{Double bubble; isoperimetric}
\communicated_by{Richard Schoen}


\setcopyrightline{Hass, Hutchings, Schlafly}

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 
$2 \pi / 3$.



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


The double bubble is the surface in $R^3$ obtained by taking two pieces of
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
size and pushes them together until they conglomerate to form a compound bubble,
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

``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.

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

Our result can also be viewed as an isoperimetric inequality.

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 $R^3$.

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


\section{An outline of the proof }

Existence and regularity of a minimizer were established by F. Almgren and J.
Almgren showed in \cite{A} that there exists an area minimizing surface in $R^3$
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
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
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

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. 

An area minimizing enclosure of $m$ volumes in $R^n$, for $ mFrom Theorem~\ref{concavity} we deduce that the volumes are connected, and hence
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
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
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
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
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}.

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



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