Paul Norbury
In the year 2000, exactly one hundred years after David Hilbert posed
his now famous list of 23 open problems, The Clay Mathematics Institute (CMI) announced its seven Millennium Problems. (http: // www.
claymath. org/ millennium ). The Gazette has asked leading Australianmathematicians to put forth their own favourite ‘Millennium Problem’.
Due to the Gazette’s limited budget, we are unfortunately not in a position to back these up with seven-figure prize monies, and have decided on
the more modest 10 Australian dollars instead.
In this issue Paul Norbury will explain his favourite open problem thatshould have made it to the list.

Lawson conjecture for minimal surfaces
The round three-sphere consists of all unit vectors in R4
x21 + x22 + x23 + x24 = 1.

(1)

As a topological manifold — it is locally the same as R3 — it is the three-sphere. Geometrically (1) has more structure since we can measure length, area and volume of subsets.
The extra structure is a metric on the three-sphere known as the round metric, so the nameround three-sphere refers to the topology and the geometry of the vectors satisfying (1) .
From the topology we know that for any closed path, or circle, there is a disk in the threesphere with boundary given by that path. The disk may have self-intersections. Plateau’sproblem asks for such a disk with smallest area. A solution to Plateau’s problem is known
as a minimal surface. More generally, a minimal surface is a surface in the three-sphere that
locally minimises area. In other words, any small circle on a minimal surface bounds a small
disk on the surface and that disk is the solution to Plateau’s problem for the small circle.
The interesting topology of the three-sphere allows compact, embedded minimal surfaces
without boundary. An example is given by any equator, also known as a great two-sphere.
Thus, the claim is that the minimal area surface that bounds a small circle on a great twosphere lives in the two-sphere. It is counter-intuitive if you think of two-spheres in R3 sincea plane in R3 that meets a two-sphere in a small circle contains a smaller area disk than thedisk on the two-sphere. This is where we see the round nature of the three-sphere, comparedto the flatness of R3 . In fact the only minimal two-spheres in the three-sphere are greattwo-spheres.
Another example of an embedded minimal surface in the round sphere is the torus
1= x23 + x24(2)2
which is known as a Clifford torus. Just as there are many equators — in fact a threedimensional space of them — there is a four-dimensional space of Clifford tori, obtained
from (2) by acting on R4 with the six-dimensional group O(4) that preserves the threesphere and its metric, and moves Clifford tori.
x21 + x22 =The 12th problem

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Lawson Conjecture. Any embedded minimal torus in the round three-sphere is a Clifford torus.
The embeddedness in the statement of the conjecture is important since there are infinitely
many immersed minimal tori [2]. The conjecture was posed by Lawson 35 years ago out of
his construction of minimal surfaces in the round three-sphere [3]. It was published as aconjecture in Yau’s survey [7] and the problems section of that proceedings.
While mentioning the age of the Lawson conjecture, let us compare it to the ages of theClay millenium problems. Thirty-five years is the age of the youngest millenium problem— the P vs NP problem — if we do not count the Yang-Mills and Mass Gap problem which
was essentially formulated when the seven problems were announced. The oldest millenium
problems are the Riemann Hypothesis — 145 years — and the Poincare conjecture — 100years — the latter which has most likely now been solved.
An inviting aspect of this problem, and more generally the study of minimal surfaces inthe round three-sphere, is the different approaches that it allows (although one may feelthat an unsolved problem has thus far allowed little.) It can be studied using PDEs, Morsetheory, Riemann surfaces and smooth topology.
PDEs arise because, as one might expect, the notion of locally minimising area can beexpressed infinitesimally, or in other words in terms of derivatives at each point of thesurface. (We shift our view slightly and think of a surface mapping to the three-sphereinstead of simply living inside the three-sphere.) From this perspective a minimal surface isa surface with vanishing mean curvature, and the mean curvature — given by the sum of
the principle curvatures — is a second order non-linear PDE satisfied by the map. See forexample [6] where you can also read about the related Willmore conjecture.
Morse theory relates the topology of a manifold and the critical points of a function on
the manifold. One might deduce topological information about a manifold from more easily
obtained critical point information, or alternatively the information may go the other way.
Consider the infinite-dimensional space of embedded tori in the round three-sphere, and afunction on this space given by area. Minimal surfaces are the critical points of the areafunction and the existence of minimal surfaces and their local properties can be deduced
from knowledge of the topology the space of embedded tori which is studied using homotopy
theory. See for example [4].
Hitchin [1] showed that harmonic maps of the torus to the round three-sphere, of which
minimal tori are examples, can be studied in terms of hyperelliptic Riemann surfaces. In
fact, a harmonic map can be completely determined by the hyperelliptic Riemann surface
equipped with a line bundle, two meromorphic differentials and constraints on their periods— objects from the study of classical Riemann surfaces. The Lawson conjecture is provenwhen the genus of the hyperelliptic curve is less than 2 and greater than 5.
Any two tangent minimal surfaces must meet at a saddle point. This is known as themaximum principle for minimal surfaces. It uses PDEs, but once the result is known, one
can work without PDEs. Thus, if topology is more to your taste than geometry, one cansimply study smooth embedded tori in the round three-sphere that meet the great spheres
and the Clifford tori in saddle points, without mention of vanishing mean curvature. Thetangent plane at any point of an embedded minimal torus is tangent to a unique great twosphere, and hence saddle to that sphere, and tangent to a one-dimensional family of Clifford
tori, and also saddle to each of them. For an example of such arguments see [5].
The Clifford torus is flat since it is (geometrically) the product of two circles. The Lawson
conjecture is proven for flat tori, where we mean to take the induced metric on the embedded

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torus. It has also been proven in other special cases. It is proven when the minimal embedded
torus has extra symmetry, say it is symmetric with respect to a circle action, or various finitegroups, and it follows from an affirmative answer to a conjecture of Yau relating the first
eigenfunction of the Laplacian on the minimal surface and its embedding.

References[1] N.J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), 627–710.
[2] W.-Y. Hsiang and H.B. Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geom.
5 (1971), 1–38.
[3] H.B. Lawson Jr., Complete minimal surfaces in S3, Ann. of Math. (2) 92 (1970), 335–374.
[4] J.T. Pitts and J.H. Rubinstein, Applications of minimax to minimal surfaces and the topology of 3manifolds, Miniconference on geometry and partial differential equations, 2 (Canberra, 1986), 137–170,
Proc. Centre Math. Anal. Aust. Nat. Univ., 12, Aust. Nat. Univ., Canberra, 1987.
[5] A. Ros, A two-piece property for compact minimal surfaces in a three-sphere, Indiana Univ. Math. J.
44 (1995), 841–849.
[6] T.J. Willmore, Riemannian geometry (Oxford University Press New York 1993).
[7] S.T. Yau, Survey on partial differential equations in differential geometry, Seminar on differential
geometry, Annals of Math Studies 102 (1982), 3-71.