Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986
Recent Progress on the Classical
Problem of Plateau
A. J. TROMBA
Introduction. In 1931 Jesse Douglas and, simultaneously, Tibor Rado solved
the famous problem of Plateau, namely, that every Jordan wire in Rn bounds at
least one disc-type surface of least area. For this work Douglas was one of the
two first Fields medalists in 1936 (the other was Lars Ahlfors). By this time he
had shown that his methods would allow one to prove that there exist minimal
surfaces of genus zero and connectivity k spanning k Jordan curves Ti,... ,Tk
in R n provided that one such surface exists having strictly less area than the
infimum of the areas of all disconnected genus zero surfaces spanning Ti,... ,Tfc
(see Figure 1). Somewhat later he announced and published proofs of theorems
giving similar sufficient but essentially unverifiable conditions that guarantee the
existence of a minimal surface of nonzero genus spanning one or more wires in
Euclidean space. The ideas of Douglas, being of great historical significance,
deserve some description and we shall begin with an analytic formulation of the
problem.
FIGURE l
© 1987 International Congress of Mathematicians 1986
1133
1134
A. J. TROMBA
Let T be a Jordan curve in R n and D C R n the closed unit disc. The classical
problem of Plateau asks that we minimize the area integral
A(u) = /
y/EG-F^dxdy
among all differentiable mappings u : D —> R n such that
u: 3D —> T is a homeomorphism.
(1)
Here we have used the traditional abbreviations
duk duk
. - £ ( £ ) • •
o - ± f f i \
, - ± dx dy '
The Euler equations of this variational problem form a system of nonlinear
partial differential equations expressing the condition that the surface u have
mean curvature zero, i.e., it is a minimal surface. One may, however, try to take
advantage of the fact that the area integral is invariant under the diffeomorphism
group of the disc and to transform these equations into a particularly simple form
by using special coordinate representations. Following Riemann, Weierstrass, H.
A. Schwarz, and Darboux one introduces isothermal coordinates
E = G,
F = 0,
(2)
which in fact linearize the Euler equations of least area; namely, they reduce to
Laplace's equation
Aw = 0.
(3)
One is thus led to the definition of a classical disc-type minimal surface as a map
u: D - • Rn that fulfills conditions (2) and (3).
For unknotted curves Gamier was able to prove the existence of solutions of
(2) and (3) subject to the boundary condition (1) by function-theoretic methods.
The general case evaded researchers until the work of Douglas and Rado. They
both used the direct method of the calculus of variations and thus obtained
and area-minimizing solution, while Garnier's solution might be unstable. In
applying the direct method one now replaces the complicated area functional by
the simpler Dirichlet functional D where
D(u) = ±f(E + G)dxdy.
It is important to note that
A(u) < H/ÊGdxdy
< ^ f(E + G)dxdy = D(u)
with equality holding if and only if E = G, F = 0. This and the analogy with
the length and energy functionals of geodesies make it plausible that minima of
D should be minima of A. This is, in fact, the case. In his prize-winning paper
however, Douglas did not explicitly attempt to find a minimum for Dirichlet's
integral but another functional H, which is now called the Douglas functional.
THE CLASSICAL PROBLEM OF PLATEAU
1135
Using Poisson's integral formula for harmonic functions, Douglas obtained the
expression
1 f22lr
* f2n
f (n(cos a, sin a) - U{COB ß, Bin ß))2 da dß
4sin 2 2V
±(a- / 0)
which equals D(u) if u is harmonic. The replacement of D(u) by H(u) transforms
a variational problem involving derivatives to one that does not, an important
feature of Douglas's existence proof. In the case of two contours Ti and Y2 in R n ,
where the domain of our mappings is an annulus, the functional H(u) is similar.
However, in the general case of surfaces of connectivity fc and genus p > 0, one
is forced to take as parameter domain a Riemann surface of genus p bounded by
fc circles, and the construction of H(u) becomes not only less elementary, but
from our point of view incredibly complicated. Douglas was able to accomplish
this generalization by making essential use of the theory of Abelian functions on
Riemann surfaces, the theory of theta functions defined on their J acobi varieties,
and their dependence on the moduli of the underlying Riemann surfaces. Namely,
in order to obtain minimal surfaces through the minimization of D or H, it is
necessary to minimize over all conformai classes of Riemann surfaces. This was
carried out by Douglas at a point in mathematical history1 when the structure
of such conformai classes was not understood. That Douglas's work was a tour
de force of classical function theory is an understatement.
According to Constance Reid's book Courant, Douglas gave a lecture at New
York University in 1936 which stimulated Courant's interest in Plateau's problem
and its generalizations to higher topological structure.
Roughly at about the same time that Courant became interested in Plateau's
problem, Marston Morse also took up the problem. He had already successfully
developed what we now call a Morse theory for geodesies and was attempting
to extend these ideas to variational problems in more than one variable. This
program was never truly successful. Plateau's problem for disc minimal surfaces
provided an ideal and pleasing test case for such a theory. Unfortunately only
very partial results were obtained and only in the disc case. Morse did not have
the notion of differentiable critical point for Plateau's problem and consequently
could never speak of the first nor the second variation of energy. Moreover he
had no idea of how to extend his generic nondegeneracy results for geodesies to
any multivariable situation and had no idea whether or not he could ever have a
finite number of solutions in any situation other than where the solutions were
known not to be unique.
Nevertheless, as he had done on other occasions he built in a definition in
order to get some result out. He defined the notion of homotopy critical point in
such a way that if you pass a critical level the topology would change as if you
had a true critical point of some finite index. Assuming only finitely many such
homotopy critical points, one could prove the existence of Morse inequalities.
*It is interesting to note that Teichmüller's pioneering work was appearing at about the
same time.
1136
A. J. TROMBA
We do not mean to imply that he had no success at all. To the contrary, he and
Tompkins and independently Shiftman showed that the existence of two disc
minimal surfaces u\ and U2 spanning a given contour, both of which provide
a strict relative minimum for Dirichlet's functional D, imply the existence of a
third disc minimal surface. This was also the first time that the "mountain pass
lemma" so popular in nonlinear analysis appeared.
In §3 we shall see that the Morse theory for disc minimal surfaces has now been
successfully completed. In addition, questions offiniteness,the existence of first
and second variations, and the nondegeneracy of solutions, all left unresolved by
early workers, have now been successfully answered. We consider these in §2.
These results are a consequence of a line of attack taken in the recent development of the classical variational approach to minimal surfaces by the author,
Reinhold Böhme, and Friedrich Tomi. The honor of giving this talk belongs
equally to them.
Finally we wish to make some remarks on the relation of the classical theory of Plateau's problem to geometric measure theory. This theory was mainly
designed to attack the higher-dimensional form of Plateau's problem, a realm
inaccessible to the classical theory. But, admittedly, also in the classical case of
two-dimensional surfaces in R 3 , the geometric measure theory approach yields
beautiful results which could not easily—if at all—be obtained within the classical theory, like the following one (due to Bob Hardt and Leon Simon): any
sufficiently smooth Jordan curve in R 3 spans an embedded (up to the boundary) minimal surface of some (unknown) topological type. Geometric measure
theory in our opinion is, however, not well suited to questions where one is interested in surfaces of a prescribed topological type. We are therefore convinced
that the classical theory continues to hold its place within the general theories
of minimal surfaces.
1. Formulation of the problem. For the purposes of exposition we shall
only consider the case of one boundary contour T C R n , bounding only oriented
surfaces. Let a: S1 —• R n be a smooth embedding with smoothness in the
Sobolev class Hr,r>7.
Let T = »(S 1 ) and let M be a Riemann surface of
genus p with dM diffeomorphic to 5 1 via some map ß: S1 —» dM. Let Ma(p) be
the <7 r ~ s_1 manifold of üfa+1 maps u of M into Rn taking dM to V and such
that u o ß is homotopic to a. Let M be the space of C°° metrics on M having
dM as a geodesic. We define Dirichlet's functional Da : M x Ma(p) —• R by
I
f
^^Dot{g^=^^rj^g(x)
2
Clgui^uiydfig
41).
i=iJM
where u = (u1,... ,un), Vgu% is the gradient of u% w.r.t. the metric g, and
d/j,g is the volume measure associated to g. Dirichlet's functional is conformally
invariant, which means the following.
Let À be any positive function on M. Then it is easy to see that
DQ(Xg,u) = Da(g,u).
(2)
THE CLASSICAL PROBLEM OF PLATEAU
1137
Moreover, we can take the Riemann surface M and form its double 2M by
gluing an exact copy M along dM. Each point z of M has an associated conjugate point z G M. The double 2M has a complex structure with the property
that the map S: z —» z is antiholomorphic. The metric g G M then extends to
a metric ga on 2M which is symmetric in the sense that S is an isometry for g3.
Let / : 2M ^ b e a symmetric G°° diffeomorphism (S*f = foS = f). Then we
also have the invariance
Pa(f*g>u°f)=Da{g,u).
(3)
Let P be the space of C°° positive symmetric real-valued functions on 2M,
and let Vo be those C°° symmetric diffeomorphisms that are homotopic to the
identity. Then (2) and (3) imply that Dirichlet's functional can be thought of as
a map
D*-T(p)xMa(p)-+R,
where T(p) = M/P/Do is the quotient space of metrics on M factored out by the
action of P and Po- This is precisely Teichmüller's moduli space, which carries
naturally the structure of a C°° smooth finite-dimensional manifold.
Since Teichmüller's space for a disc Z) is a single point, it follows that if M = D
expression (1) reduces to the classically known expression for Da, namely,
The basic result (originally established in a totally different context by
Douglas) is that the critical points of Da\ T(p) x Na(p) —• R consist of pairs
(TO) UO), where TQ represents a conformai equivalence class of metrics (or equivalente a complex structure on M) and a conformai map UQ: M —• R n ; i.e.,
u
o9n = Aö'OJ where gn is the R Euclidean metric and go is any metric representative in the conformai class represented by To.
For each a, at least for each a which is real analytic, we would like to be able
to say that the number of critical points for P a is finite. This problem is, as yet,
unsolved. However, in the case M — D, the answer is known in the generic case.
This is the subject of our next section.
2. The index theorem for disc surfaces. If M = D then, as remarked
earlier, Dirichlet's functional takes the form Da : Ma —• .#, where
-ti
0«(«) = £ Ê / " Vu*-Vu*
and for p = 0 we denote Ma(p) simply by Ma.
Let us begin by asking the question of determining the structure of the set
of all minimal surfaces (area-minimizing or not). In this direction let A be the
open set (in Hr) of all embeddings a, and M = \Ja Ma. Then M is a smooth
fibre bundle over A with projection map 7r: M —* A given by w(u) = a if u G Na.
Let E c i / denote the set of all minimal surfaces.
1138
A. J. TROMBA
One might, at first glance, conjecture that E is a submanifold as would be
the case for the geodesic problem with fixed endpoints. Surprisingly E is not a
manifold but an "infinite-dimensional algebraic variety" composed of manifold
strata determined by the singularity structure of the minimal surface. To be
more precise let u : D —> R n be a minimal surface. Then F = du/dx — idu/dy
is a holomorphic map of D into C n . A point ZQ where u fails to be an immersion
is a zero for F. We say that zo G D is a branch point of u of order A if F(z) =
(z — zo)xG(z), where G(zo) ^ 0. Clearly every interior branch point has some
finite order. If a is sufficiently smooth, it can be shown that all boundary branch
points have a finite order.
DEFINITION. Let A G Zp and v G Zq, X = ( À i , . . . , Ap), v = ( i / x , . . . , vq), be
tuples of integers. We say that a minimal surface u has branching type (A, v)
if u has p (arbitrarily located) interior branch points z±,..., zp G D° of orders
A i , . . . , Ap and q boundary branch points £ 1 , . . . , £q of orders v\,..., vq.
Let E* denote all the minimal surfaces in the bundle M of branching type
(A,^). Let 7r£ = 7T|E£ be the restriction of the bundle projection map to E*.
The following is then known as the Index Theorem for Minimal Surfaces [1].
THEOREM. Each S£ is a manifold with Eg a submanifold of M. The map
7T* : E* —y A is Fredholm of index
I(X, i/) = 2|A|(2 - n) + |i/|(2 - n) + 2p + q + 3, 2
where |A| = E* ; \v\ = E ^ .
This index result is the basis of proving the generic (open-dense) finiteness
and stability of minimal surfaces of disc type. The open dense set will be the
set of regular values for the map 7TE = 7r|E. In addition, this theorem leads us
to a new definition of nondegeneracy in critical point theory. A minimal surface
u G E* is nondegenerate if I(X, v) — 3 (and hence v — 0) and the map 7TQ
restricted to a neighborhood of u in EQ is a local diffeomorphism.
If n > 3 the nondegenerate minimal surfaces are immersed up to the boundary and if n = 3 they are either immersed or simply branched (A = ( 1 , . . . , 1)).
We should re-emphasize, at this point, that we are considering not only areaminimizing minimal surfaces bounded by a G A but all critical points. There
are some additional, surprising consequences of this index formula. First (by
Gulliver-Ossermann and Alt), area-minimizing minimal surfaces in R 3 are free
of interior branch points, whereas most minimal surfaces which are either immersed or have simple interior branch points are stable w.r.t. perturbations of
^the^oundaryr^Second^for^r^^>=4-minimal^surfaces^in^Rû=may^have=branGh==
points even if they are area-minimizing but for such n no minimal surface in R n
is stable under perturbations of the boundary.
This index theorem has been generalized to fc-connected regions by Karl
Schüffier and Ursula Thiel [4, 6]. Recently Gulliver and Hildebrandt [2] have
2
The number 3 arises from the conformai invariance of the problem under the action of the
3-dimensional conformai group of the disc. The number 3 would disappear if, for example, we
imposed a three-point condition on our maps u.
THE CLASSICAL PROBLEM OF PLATEAU
1139
given a beautiful example of three coaxial circles that bound a continuum of
2-connected minimal surfaces of genus zero.
3. Morse theory. As a consequence of the index theorem we know that the
generic curve a in R 4 bounds only a finite number ui,..., um of disc minimal
surfaces. For each Ui one can show that there is a finite Morse index 0{ for the
Hessian of P a at U{. The integer 9i is the dimension of the largest subspace on
which D2Da(ui) is negative definite. If a complete Morse theory held for this
problem (a handle body decomposition of Ma in terms of the critical points of
Da) one would have, in particular, the Morse equality
£(-l)"' = L
(!)
i
This equality was established by the author [8, 9].
The difficulty with directly applying the Morse-theoretic ideas of Palais and
Smale is that P a does not satisfy their connection C.
Nevertheless in a beautiful recent paper Michael Struwe [5] has shown that by
restricting Da to the "convex" set M^ c Ma of these maps u which are monotonie
on the boundary, one does have a full Morse theory for disc minimal surfaces
spanning contours in R 4 .
In R 3 the generic wire still bounds only finitely many minimal surfaces of disc
type. Moreover, as it turns out, these minimal surfaces are zeros of a Fredholm
vector field
**-Oi '• Met —* •* MOf
This permits one to define a rotation number of X a about each zero and hence
the total Euler characteristic xfàa) °f X a (which is the sum of the rotation
numbers). The corresponding theorem in R 3 is that
X(X„) = 1.
(2)
Finally, there is another interpretation of (1) in R 4 . One can show that the
main stratum E§ of E is an (infinite-dimensional) oriented manifold. This allows
one to define the degree of the map 7TE = 7r|E. Then
deg7TE = l.
(3)
4. Existence of higher genus surfaces. Let us consider the pictures of
physical soap films, as shown in Figure 2.
Although Douglas showed that any Jordan contour always bounds a disc
minimal surface, no criterion was known, until recently, on a contour T which
guaranteed the existence of a genus p > 0 minimal surface spanning T. All of
the above physical examples of soap films can now be partially 3 explained by
the following existence theorem [7] due to F. Tomi and the author.
3
We use the word partially because we do not yet know whether our solutions are embedded
as in the soap film examples.
1140
A. J. TROMBA
FIGURE 2
THEOREM. Let M be a surface of genus p > 1 with one boundary component,
and let V be a rectifiable Jordan curve in R 3 . We assume that
(i) there exists a solid q-torus T of class C 3 in R 3 with positive inward mean
curvature such that T C T, where q = 2p if M is orientable and q = p if not,
and
(ii) with respect to suitably chosen base points and generators the class of Y in
7Ti(T) is represented by the same word as the class of dM in TTI(M), respectively.
Then there exists a minimal surface f: M —• R 3 mapping dM topologically onto
T and f(M) contained in T.
The hypothesis on T is fulfilled in the following specific cases:
(i) M is oriented of genus p, and T is homotopic in T to the commutator
product n j = i [ a i » / y f° r s o m e set ai,...,ßp
of free generators of TTI(T);
(ii) M is nonorientable of genus p and V is homotopic in T to n j = i &ji where
a i , . . . , a p are free generators of 7Ti(T);
(iii) M is nonorientable of genus p — 2k + 1 and T is homotopic in T to
(rij=i [aj>ßj])l2i where a±,..., a^, 7 are free generators of 7Ti(T);
(iv) M is nonorientable of genus p = 2k, k > 1, and T is homotopic in T to
(JljZi[a3>ßA)akßk<Xklßk> f o r generators a i , . . . , / ? * of TTI(T).
5. Teichmüller theory and Plateau's problem. The difficulty in obtaining the existence of minimal surfaces of genus p > 0 spanning a given contour T in
-^Euclidian-space=by^the direct_method-of the^calculus of^var=iations4s=that one-can—
not, in general, show that a minimizing sequence (rn,un), Pof(rn,iAn) —> inf Da,
has a convergent subsequence in some reasonable topology. The problem is that a
minimizing sequence might be "degenerating" in some sense to a surface of lower
genus. For the existence theorem of the last section a way, given the hypotheses,
is found to prevent degeneration.
However if we would like to develop a general index theory or a Morse theory
covering all genera, or at least be able to prove generic finiteness, one must allow,
THE CLASSICAL PROBLEM OF PLATEAU
1141
and in fact be able to completely understand, such degeneration. This amounts
to passing through some hypothetical boundary of Teichmüller space T(p) to
another Teichmüller space T(p'), p' < p. From this point of view it would be
nice if we had a universal Teichmüller space on which to define our problem.
Such spaces have been investigated by Bers.
Since much of Plateau's problem is differential-geometric in flavor, one would
like to approach this question along the lines of differential geometry. One could
ask if there was a "natural" Riemannian structure on Teichmüller space whose
curvature could be computed. If the sectional curvature were negative and the
associated metric complete, one could then compactify along the lines developed
by Eberline and O'Niell in their theory of visibility manifolds.
A Riemannian structure on Teichmüller space originally arising in number
theory was introduced by Weil and then studied by Ahlfors, the so-called WeilPetersson metric. This metric was shown by Ahlfors to have negative Ricci and
holomorphic sectional curvature. Wolpert and Thu showed that the metric was
incomplete and therefore Eberline-O'Niell compactness would not follow should
the metric be negatively curved. However, from the point of view of minimal
surface this incompleteness rather than being a defect makes the metric even
more interesting.
Another motivation for studying the Weil-Petersson metric comes from General Relativity, where this metric has appeared in a totally different context.
We have proved the following result [10].
THEOREM.
tive.
The sectional curvature of the Weil-Petersson metric is nega-
Since this result, Wolpert [11] and then later Jost [3] have computed the
Riemann curvature tensor of this metric. It appears that the geodesies of T(p)
with respect to this metric may provide a mechanism to attain a compactification
of Teichmüller space via differential geometry.
How this will impact, if at all, the development of Morse theory, index theory,
and generic finiteness for higher genus awaits future developments.
REFERENCES
1. R. Böhme and A. J. Tromba, The index theorem for classical minimal surfaces,
Ann. of Math. (2) 113 (1981).
2. R. Gulliver and S. Hildebrandt, Boundary configurations spanning continua of minimal surfaces, Manuscripta Math. 54 (1986), 323-347.
3. J. Jost, Harmonic maps and Teichmüller theory, Preprint.
4. K. Schüffler, Eine globalanalytische Behandlung des Douglas1 sehen Problems,
Manuscripta Math. 48 (1984), 189-226.
5. M. Struwe, A critical point theory for minimal surfaces spanning a curve in R n ,
J. Reine Angew. Math. 349 (1984), 1-23.
6. U. Thiel, The index theorem for k-fold connected minimal surfaces (to appear).
7. F. Tomi and A. J. Tromba, On Plateau's problem for minimal surfaces of higher
genus in R 3 , Bull. Amer. Math. Soc. (N.S.)13 (1985), 169-171.
1142
A. J. TROMBA
8. A. J. Tromba, Degree theory on oriented infinite dimensional varieties and the
Morse number of minimal surfaces spanning a curve in R n . I: n > 4, Trans. Amer. Math.
Soc. 290 (1985), 383-413.
9.
, Degree theory on oriented infinite dimensional varieties and the Morse
number of minimal surfaces spanning a curve in R n . II, Manuscripta Math. 48 (1984),
139-161.
10.
, On a natural algebraic affine connection on the space of almost complex
structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric, Manuscripta Math, (to appear).
11. S. Wolpert, The topology and geometry of the moduli space of Riemann surface,
Arbeitstagung, Bonn, 1984; Lecture Notes in Math., vol. 1111, Springer-Verlag, Berlin and
New York, pp. 431-451.
UNIVERSITY OF CALIFORNIA, SANTA CRUZ, CALIFORNIA 95064, USA