**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.math.psu.edu/era/.
**

\documentstyle[amscd,amstex,amssymb]{era-l} \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{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 examples 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 $ mFrom 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). \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). \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). \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). \bibitem{Moore} R. E. Moore, {\em Methods and Applications of Interval Analysis,} SIAM, 1979. \bibitem{Pl} J. Plateau, {\em Statique experimentale et theorique des liquides soumis aux seules forces molecularies,} 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). \newpage \bibitem{Th} D'arcy Thompson, {\em On growth and form,} Cambridge Univ. Press, NY, 1959, (first edition 1917). \end{thebibliography} \end{document}