Constant Mean Curvature Surfaces in Euclidean Spaces
NIKOLAUS
KAPOULEAS*
Department of Mathematics, Brown University
151 Thayer Street, Box 1917
Providence, RI 02912, USA
A variant of the isoperimetric problem is to classify and study the hypersurfaces in
the Euclidean space E n + 1 that have critical area subject to the requirement that
they enclose a fixed volume. In physical terms this is equivalent to having a soap
film in equilibrium under its surface tension and a uniform gas pressure applied to
one of its sides; hence, such surfaces are often called soap bubbles. The geometric
condition for such a surface is that its mean curvature H is a nonzero constant.
The precise value of the constant is not important because it can be changed to any
desired value by a nomothetic expansion. We will be using the abbreviation "CMC
surface" to mean "complete smooth hypersurface properly immersed in E n + 1 with
H = 1". Notice that the above definitions do not require embeddedness.
Although it has been known and proven for a long time that the round
spheres are the unique answer to the isoperimetric problem, only recently has the
corresponding question been settled for soap bubbles. Actually up to 1980 the only
known examples of CMC surfaces of finite topological type were the rotationally
invariant ones in E 3 studied by Delaunay in 1841 [5]. This, combined with various
nonexistence results we will now review, led to the suspicion — sometimes called
the Hopf conjecture although it is not clear that Hopf ever took sides on this
question — that the round sphere is the only closed bubble in E 3 .
Jellet [11] had already proved in 1853 that starshaped closed CMC surfaces
are round spheres. A century later Hopf [9] proved that a CMC surface homeomorphic to S2 is a round sphere. His proof uses the so-called Hopf differential,
which is a quadratic holomorphic differential for the underlying Riemann surface
structure of the surface M defined by
* = (Xzz, i/) dz2 = {(An
- A22 - 2iA12)dz2 ,
where z = u -\- iv = x\ -\- ix2 is a local isothermal coordinate, X : M —> E 3 is
the immersion in consideration, v : M —> E 3 its Gauss map, (., . ) the standard
inner product in E 3 , and Aij the second fundamental form. The Cauchy-Riemann
equations establishing the holomorphicity amount to the Codazzi equations once
H = 1 is used. Clearly (by Liouville's theorem for example) holomorphic quadratic
*)
Partially supported by NSF grants DMS-9404657 and NYI DMS-9357616 and a Sloan
Research Fellowship.
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
482
Nikolaos Kapouleas
differentials on S2 vanish. Hence, A = Hg (g is the first fundamental form), g has
constant curvature, and X immerses to a round sphere.
In 1956 Alexandrov [2] proved that all embedded closed CMC surfaces (any
n) are round spheres. His method uses moving planes to establish by the use of
the maximum principle that there is a plane of symmetry parallel to any given
plane; hence, the surface is a round sphere. Both Hopfs and Alexandrov's methods
have found numerous applications to other problems; actually there is currently no
other method of the wide applicability of Alexandrov's in dealing with questions
of uniqueness and symmetry in nonlinear problems. More recently, Barbosa and
doCarmo [3] showed that local minimizers of the variational problem are round
spheres.
In the 1980s the general picture changed. In 1982 Hsiang [10] demonstrated
that Hopfs theorem is not valid in higher dimensions by constructing nonround
CMC spheres. His method was to reduce the problem to an ordinary differential
equation (ODE) by imposing nonstandard rotational symmetry. In 1984 Wente [27]
in a surprising development disproved the so-called Hopf conjecture by producing
toroidal soap bubbles in E 3 . Because a torus can be covered conformally by C we
can arrange that the Hopf differential lifts to <Ë> = dz2 where z = u + iv is the
standard coordinate on C. By writing then the fundamental forms as
\e2w\dz\2,
4
'
'
A = ^-r^dv?
+ ^ ± d v
A
2
A
the Gauss equation reduces to the sinh-Gordon equation
Aiu + ^sinhiu = 0,
hence solutions of this equation integrate on C to conformai CMC immersions.
Wente using partial differential equation (PDE) methods found a 2-parameter
family of highly symmetric doubly periodic solutions w and he demonstrated that
the parameters can be arranged so that the corresponding CMC immersion X is
also doubly periodic. Abresch [1] subsequently realized that one can find the solutions w Wente used by separation of variables and he demonstrated that there are
Wente tori with only 3 positively curved regions. Walter [26] expressed the immersion X in closed form using theta functions and gave a very detailed description
of the geometry of these surfaces.
Soon afterwards Pinkall and Sterling [22] classified all the doubly periodic solutions w and hence the CMC tori; this result could be thought of as the analogue
of the Enneper-Weierstrass representation for minimal surfaces. Bobenko [4] noticed the analogy with the classical soliton theory where one has the sine-Gordon
equation instead, and he improved and generalized this classification. Many other
people have been working in this direction. An interesting result, for example, of
Ercolani, Knörrer, and Trubowitz [7] has been the proof that there are continuous
families of CMC tori of arbitrarily large number of parameters. One should mention at this point the Hsiang-Lawson conjecture that the only minimal embedded
torus in S3(l) is the Clifford torus. In spite of the fact that all the attempts have
failed up to now, there is still hope that the classification of the minimal tori in
S 3 (l) [4] will help in settling this conjecture.
Constant Mean Curvature Surfaces in Euclidean Spaces
483
All the CMC surfaces of finite topological type known by the above methods
are topological spheres, cylinders, planes, tori, or tori with two ends. Attempts
to extend to other topological types have been unsuccessful up to now for two
reasons. First, umbilics always exist then. Second, the fundamental group is not
commutative as in the case of a torus and hence the induced representation into
the Euclidean group does not have to consist of Euclidean motions sharing a
common axis. Fortunately, there is another general construction [12]-[16] that gives
a plethora of examples for almost any finite topological type, including examples of
closed surfaces of any genus besides genus 0 and 1, and also embedded (complete)
examples of any genus and enough ends. Most of the remaining discussion will
concentrate on this construction.
The most ambitious formulation of the construction would be to give general
conditions under which the following is possible: start with a collection of unit
spheres in E n + 1 from each of which a number of small discs is removed, and a
collection of complete minimal surfaces rescaled to small size from which a neighborhood of infinity is removed. Span then the existing boundaries with surfaces so
that a complete surface is obtained that is subsequently perturbed to a CMC surface. Such a construction seems plausible because all known CMC surfaces come
in families containing surfaces that can be decomposed as above: small perturbations of unit spheres minus small discs, rescaled complete minimal surfaces minus a
neighborhood of oo, and regions connecting the two. The main difficulty in having
such a general construction is the construction of the connecting surfaces. At the
moment we can only borrow the connecting parts from the Delaunay or the Wente
cylinders, whose geometry we proceed to describe.
The Delaunay family of surfaces can be parametrized by a single parameter
r to be defined later. Each surface is obtained by rotating a periodic curve around
the axis, r can take both positive and negative values; for positive r the curve is
the graph of a function and the surface is embedded, whereas for negative r it is
not. Actually, Delaunay produced these curves as the loci of a focus of an ellipse
(r > 0) or a hyperbola (r < 0) rolling on the axis. {K = 0} on the surface (K
denotes the Gauss curvature) is a union of circles whose removal disconnects it into
components that we call almost spherical regions (asr's for short) for reasons that
will become clear later. We call an asr positive or negative according to the sign of
K on it. If \T\ is small a positive asr minus a small neighborhood of its boundary
approaches a round sphere of radius 1 — recall H has been normalized to be 1
and we assume the conventions that give H = 1 on Sn(l) — minus two small
antipodal discs. Similarly, an enlarged negative asr by a factor of | r | _ 1 suitably
placed approaches a catenoid {x2 -\- x2 — cosh 2 a;i} on any large ball centered
at the origin of the coordinate system Ox\x2x^ in consideration. In the limit as
r —> 0 the surface becomes a string of unit spheres touching externally and with
centers on the axis while the negative asr's shrink to points. The rescaled as above
negative asr's tend to catenoids.
There is a translation along the axis of the Delaunay surface that carries a
positive asr to one that is separated from the previous one by a single negative
asr. The length of this translation is an increasing function of r, for r = 0 being
2. Finally, to define r precisely, consider the corresponding physical system of
484
Nikolaos Kapouleas
soap film and enclosed gas in pressure. If one separates it into two components
by cutting with a surface, then the part on the left exerts a force on the part on
the right, r is so defined that this force is 7rre, where e is the unit vector in the
direction of the axis of the Delaunay surface pointing from right to left. Notice that
this force is attractive or repulsive according to the sign of r and does not depend
on the cutting surface because the (finite) part of the system isolated between two
different cuts has to be in equilibrium.
We turn our attention now to the Wente surfaces. By suitably choosing one
of the free parameters of the Wente construction it is possible to arrange for the
immersion X : C —• E 3 to have a period. This way we obtain a 1-parameter family
of Wente cylinders. We call the parameter r again although we do not have a
nice definition for r as before. In this case r is restricted to positive values. The
Wente cylinders like the Delaunay ones are periodic, the period now is however
a rotation around an axis instead of a translation. The angle of this rotation 9
varies continuously with r and 9 —> 0 as r —» 0. The Wente cylinders close up to
tori whenever 9/-K is rational. There is a plane of symmetry P of the whole surface
that is perpendicular to the axis.
We define asr's as before, the only difference being that {K = 0} is a highly
connected union of curves now. For small r each positive asr — excluding as usual
a small neighborhood of its boundary — approaches a unit sphere minus one
small disc. Each negative asr enlarged by a factor of | r | _ 1 approaches similarly
an Enneper's minimal surface. Asr's come in pairs, each pair contains one positive
and one negative asr and their common boundary. The boundary of each pair
consists of two circles, each of which is a generator of the fundamental group of
the cylinder. Each of these circles immerses as a planar "figure 8" of maximum
symmetry. Its plane is parallel to the axis at distance 1/2. Successive planes form
an angle of 9/2. The bisector of the angle of two successive planes is a plane of
symmetry of the pair in between, and so is P. The self-intersection q of each figure
8 is the projection on the plane of the figure 8 of the intersection p of the axis with
P, each line pq is a line of reflectional symmetry of the whole cylinder. Using such
reflections the whole cylinder is generated by a single pair of asr's.
An important difference from the Delaunay case, besides the complicated
nature of {K = 0}, is the fact that each asr is attached to the rest of the surface
at one place only. Hence, the force exerted on it by the rest of the physical system
has to vanish, whereas in the Delaunay case we have two nonzero forces exerted,
one at each of the two antipodal attachments, canceling each other. We could
consider the two forces exerted on a pair of asr's (we have two attachments then),
but these also vanish.
We go back now to discussing the construction of new CMC surfaces. At
the present stage of development of this construction we are restricted to the
following approach: fuse positive asr's from Delaunay or Wente cylinders or tori
and unit spheres, thereby creating a CMC surface M on which H — 1 is supported
on some of the positive asr's — called central in the rest of the discussion. Find
then a function <ß : M —> E such that X^ := X + <f>v is a CMC immersion (X
and v are the initial immersion and its Gauss map)'. Examples of such M's are
a sphere to which a number of Delaunay ends has been attached, two spheres
Constant Mean Curvature Surfaces in Euclidean Spaces
485
connected by a Delaunay neck, or two Wente tori placed symmetrically so that
two particular positive asr's, one from each, are fused together so that a genus
2 closed surface is obtained. It is clear from the second example that there must
exist an obstruction to the construction because otherwise we would contradict
both Hopfs and Alexandrov's theorems.
The idea for finding cj) is to use perturbation methods, based on the fact that
if the r parameters of the Delaunay and Wente ingredients are small, then H — 1
is small (actually of order r ) . We would like to linearize and correct for H — 1,
and then correct for the higher order terms. This can be phrased in the language
of fixed point theorems. Let H^ be the mean curvature of X^, then
i ^ = tf+±£0 + Q0,
where £ = A + \A\2, \A\ is the length of the second fundamental form, and Q^
is quadratic and higher order in cj) with geometric invariants of X as coefficients.
Given cj), let u, v be the solutions to the linear equations
Cu = 2(1 - H),
£v = 2Q(f}.
Clearly then, a fixed point of the map cj) —> u — v provides us with the desired cj).
Our approach hence requires us to be able to invert £ and to be able to
have good enough estimates for the solutions and the higher order terms. The
higher order terms in particular create various technical difficulties because the
coefficients blow up as r —> 0 on the negative asr's. The main difficulty however is
that it is not clear that £ is even invertible. Actually on the unit spheres and the
complete minimal surfaces one obtains in the limit as r —> 0, £ is not invertible
because the translations give rise to a 3-dimensional kernel.
In order to resolve these difficulties it is helpful to employ the conformai
covariance of the Laplacian in dimension 2 to appropriately change the equations
and facilitate their study. By changing the metric to h — \\A^g the linearized
equation changes to £u = 4|A|~ 2 (1 — H), where £)x \— A/t -J- 2, while the metric
on the positive asr's changes very little and on the negative asr's changes to make
them isometric to the positive asr's. £]x has then 3 small eigenvalues for each asr
of M corresponding to the 3-dimensional space of translations. We call the space
spanned by the corresponding eigenfunctions an approximate kernel. Moreover,
the components of the Gauss map on each asr are functions close in L2 to this
approximate kernel. Hence, there is no hope of bringing our construction to a
successful conclusion unless |A| - 2 (1 — H) is (approximately) orthogonal to these
functions. This is the obstruction we have been expecting and by using the physical
model one can see it amounts to the requirement that the forces exerted on each
central asr by the rest of the system have vanishing resultant. This is identical —
but in the language of physics — to applying the balancing formula in [18], where
such balancing arguments were first developed for CMC surfaces.
As we have already seen, each Delaunay piece attached to a central asr exerts
a force 7rre, where e is in the direction of the axis of the Delaunay and points
away from the asr towards the piece. The Wente pieces do not contribute to the
resultant force. We assume from now on that the balancing condition is satisfied
486
Nikolaos Kapouleas
by M; that is, for each central asr we have ^ i~e = 0, the summation being over
all the Delaunay pieces attached to the asr in consideration. We can then solve the
linear equations in our scheme modulo small elements of the approximate kernel.
Assuming the required weighted estimates for the smallness of Q^, we can ensure
the existence of a 0 such that H^ — 1 is a small element of the approximate kernel.
It remains to correct for this approximate kernel. To this purpose we introduce some further perturbation of M that will introduce H — 1 with approximate
kernel content canceling the one we want to remove. If there are only Delaunay surfaces used in the construction, we can prescribe any small element of approximate
kernel by perturbing M as follows: prescribing the projection to the approximate
kernel of H — 1 is approximately the same as prescribing fs(H — 1)Uì on each asr
S, where the zVs are the components of the Gauss map. This is in turn the zth
component of the resultant force exerted on S by the rest of the system. In the
Delaunay case the rest of the system has two components; thus, we can create a
resultant by changing the parameters of one of them (r or the direction). The force
then exerted by this Delaunay piece does not balance anymore the force exerted
by the other.
This approach clearly fails when there are Wente surfaces incorporated in
the construction. One needs again to be able to prescribe a resultant force to each
asr of the Wente surface, but now there is only one component on the rest of the
system and this exerts no force for any value of the parameter. Nevertheless, it
is still possible to create such a force as follows: excise most of each asr from the
Wente surface to leave a small neighborhood N of {K = 0}. The first Dirichlet
eigenvalue of £h on N is (large) positive and N is stable. Thinking of the boundary
components of N as wires we can move them to new positions without destroying
the soap film; this repositioning should be resisted by forces trying to bring the
wires to their original positions. By reattaching the excised parts of the asr's at
the new positions we should have been successful in creating |A| - 2 (i7 — 1) with
approximate kernel content.
This is indeed so and one way to make it precise is as follows: Let 7 b e a
Killing vector field of the ambient E 3 . Clearly then £h(v -Y) = 0 . Hence, if / is
some function on some domain ft of M we have using integration by parts that
/ dphv-Y
Jn
Chf=
[
Jon
V(f)vY-fV(vY),
where V is the outward unit normal to dfl tangent to M. If we ignore higher order
terms the left-hand side is the projection to v • Y of the | A | _ 2 ( i î — 1) produced
by changing X to Xf. For example, suppose that we construct / on the Wente
cylinder W as follows: solve the Dirichlet problem 4 / = 0 on JV (as above) with
/ = 0 on dû \ S and / = u • e on dN (1 S, where S is an asr and e is a unit vector
parallel to the intersection of the two planes of symmetry of S. Extend / to be
v • e on S \ N and 0 on the rest of W. Finally smooth it out by changing it on
a neighborhood of dN. Clearly then, the left-hand side above with D, = S and
Y = e is up to a constant the approximate kernel created corresponding to e on S,
whereas the right-hand side can be calculated to be of order | log r | _ 1 by arguing as
Constant Mean Curvature Surfaces in Euclidean Spaces
487
follows: / is approximately harmonic on N. Most of SnN is conformally isometric
to a cylinder S1(l)x[0,ß], where E « | | logr|. Because harmonicity is conformally
invariant, / œ 1 on dNnS, and — as it turns out — / « 0 on 9 5 , we see that the
flux of / through the generator of ivi(S n N) is of order | logT| _ 1 . In this case the
flux is the dominant term on the right-hand side and the argument is complete.
This general approach of creating approximate kernels can be summarized by what
we call the Geometric Principle:
Creation of (H - 1)|^4|~2 in the approximate kernel direction amounts to repositioning the asr's reiati ve to each other.
GEOMETRIC PRINCIPLE.
Notice that in the earlier approach in the Delaunay case this is definitely
valid because changing the parameter (and hence the period) or the direction of
the axis clearly repositions the asr's. It is conceptually illuminating also to notice
that for the above / we have essentially proven that £]%f has a projection to one
of the eigenfunctions of the approximate kernel of order | l o g r | - 1 . Because / has
a projection of order 1, the corresponding eigenvalue is also of order | l o g r | _ 1 . It
turns out that the small eigenvalues of £]x in the fusion of Wente tori are of two
kinds: those of order | l o g r | _ 1 , and those of order y/r. From all this we can conclude that our approach amounts to effectively inverting the linearized operator in
the direction of the approximate kernel. This method and the "geometric principle" should be widely applicable in all similar problems where small but nonzero
eigenvalues appear.
Although the above gives the philosophy of how to prescribe the creation
of the kernel in the Wente case, much more work is required to actually do so.
The difficulty is due to the fact that in a short distance from an asr there are
many (their number tends to oo as r —> 0) asr's. One succeeds by using the
precise information one has for v • Y on the figure 8's separating pairs of asr's, the
symmetries of the Wente cylinders, the approximate harmonicity of the various / ' s
as above, the conformai invariance of the harmonic condition, and various other
ideas for which we refer the reader to [16]. We only remark here that it fits with the
rotational character of the Wente cylinders that in the end one has to prescribe not
only forces through the figure 8's but also torques. These are created by a relative
rotation of the two components of the complement of the figure 8 in consideration
around the axis of the cylinder.
We would like to make some final comments concerning the construction of
closed surfaces by this method. If one uses Delaunay surfaces only, then one has
to have a number of central spheres. Because of the balancing condition, at least
3 Delaunay pieces have to emanate from each central sphere; hence, there are
at least 4 central spheres and the genus is at least 3. To satisfy the balancing
condition at an outermost central sphere one has to use Delaunay pieces of both
the embedded and nonembedded kind so that some of the forces are repulsive and
some attractive. In this case there is another difficulty [14] as well: once all but
one of the Delaunay pieces are placed, one has to place a final one to connect
central spheres whose positions are fixed already. In general this last piece will not
fit. This difficulty can be overcome by using a free parameter one has available to
adjust the distance of the two spheres to that required by the piece to fit in. This
488
Nikolaos Kapouleas
requires the use of very long Delaunay pieces, which magnifies this effect. Now
using Wente cylinders one can produce closed surfaces of genus 2 as well, as in the
example we have already mentioned. Here again we have a period problem, which
comes into the construction in a subtler fashion; for details see [16].
In the case of complete embedded surfaces we can only use Delaunay pieces
connecting central spheres and Delaunay ends. The ends of the surfaces we obtain
decay exponentially to the Delaunay ends at oo. Korevaar, Kusner, and Solomon
[19], extending partial results of Meeks [21], proved that this is the case in general;
that is, every embedded CMC surface in E 3 has ends that decay exponentially
at oo to Delaunay ends. More recently, Korevaar and Kusner [18] have further
restricted the structure of a general embedded CMC surface to resemble more the
structure of the ones constructed as above. Concerning the general structure an
extra piece of information comes from a construction of Grosse-Brauckmann [8].
Extending methods of Lawson [20] and Karcher [17] he constructed a 1-parameter
continuous family of CMC spheres of maximum symmetry with n ends (n > 3).
The family starts with surfaces like the ones already discussed, where n embedded
Delaunay ends are attached to a central sphere. The parameter r then of the ends
increases to a maximum value and then decreases again towards 0. These last
surfaces have n Delaunay ends of small r > 0 joined in the middle by an n-noid.
In summary we can say that in the last few years there has been enormous
progress in the subject. We now understand that there is a rich variety of CMC
surfaces and we have made substantial progress in understanding the subject in
its totality. Still, there are many unanswered questions and some which come to
mind are the following:
(1) Very little is known in higher dimensions.
(2) Classify the connected components of the moduli. In the embedded case a lot
of progress has been made [19]. In the immersed case the statement should be
modified somehow so that all Wente tori count as being in the same component,
for example.
(3) Understand better the structure of each component in the spirit of [8]. Perhaps
a more general construction based on minimax methods would be useful here.
(4) As an introductory step, understand the geometry of the CMC tori; in particular, what kind of minimal surfaces appear in their degenerations.
Finally, we mention that perturbation methods have been used in other geometric problems, for example with great success for instantons [17], [6] and mininal
surfaces [24]. Closer in spirit are constant scalar curvature constructions [23]. There
are other proposed constructions, for example for minimal surfaces or Einstein 4manifolds, which in some respects are very similar to the one we discussed and
which are still open.
Constant Mean Curvature Surfaces in Euclidean Spaces
489
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[io;
[n;
[12;
[13;
[14;
[15:
[ie:
[17:
[18
[19
[20:
[21
[22
U. Abresch, Constant mean curvature tori in terms of elliptic functions, J. Reine
Angew. Math. 374 (1987), 169-192.
A. D. Alexandrov, Uniqueness theorems for surfaces in the large I, Vestnik Leningrad Univ. Math. 11 (1956), 5-17.
L. Barbosa and M. doCarmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), 339-353.
A. I. Bobenko, All constant mean curvature tori in R , S*3, H3 in terms of thetafunctions, Math. Ann. 290 (1991), 209-245.
D. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante,
J. Math. Pures Appi. (4) 6 (1841), 309-320.
S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-manifolds, Clarendon Press, Oxford, 1990.
Nick Ercolani, Horst Knörrer, and Eugene Trubowitz, Hyperelliptic curves that
generate constant mean curvature tori in R , The Verdier Memorial Conference
on Integrable Systems: Actes du colloque international de Luminy (1991), ed. by
Olivier Babelon, Pierre Cartier, and Yvette Kosmann-Schwarzbach, Birkhäuser,
Basel and Boston.
K. Grosse-Brauckmann, New surfaces of constant mean curvature, Math. Z. 214
(1993), 527-565.
H. Hopf, Lectures on differential geometry in the large, Notes by John Gray, Stanford University, Stanford, CA.
Wu-Yi Hsiang, Generalized rotational hypersurfaces of constant mean curvature in
the Euclidean spaces, I., J. Differential Geom. 17 (1982), 337-356.
J. H. Jellet, Sur la surface dont la coubure moyenne est constante, J. Math. Pures
Appi. (9) 18 (1853), 163-167.
N. Kapouleas, Constant mean curvature surfaces in Euclidean three-space, Bull.
Amer. Math. Soc. 17, (1987), 318-320.
, Complete constant mean curvature surfaces in Euclidean three-space, Ann.
of Math. (2) 131 (1990), 239-330.
, Compact constant mean curvature surfaces in Euclidean three-space, J.
Differential Geom. 33 (1991), 683-715.
, Constant mean curvature surfaces constructed by fusing Wente tori, Proc.
Nat. Acad. Sci. USA 89 (1992), 5695-5698.
, Constant mean curvature surfaces constructed by fusing Wente tori, Invent.
Math., to appear.
H. Karcher, The triply periodic minimal surfaces of A. Schoen and their constant
mean curvature companions, Man. Math. 64 (1989), 291-357.
N. Korevaar and R. Kusner, The structure of complete embedded surfaces with
constant mean curvature, J. Differential Geom. 30 (1989).
N. Korevaar, R. Kusner, and B. Solomon, The structure of complete embedded
surfaces with constant mean curvature, J. Differential Geom. 30 (1989), 465-503.
H. B Lawson, Jr., Complete minimal surfaces in S , Ann. of Math. (2) 90 (1970),
335-374.
W. H. Meeks, III, The topology and geometry of embedded surfaces of constant mean
curvature, J. Differential Geom. 27 (1988), 539-552.
U. Pinkall and I. Sterling, On the classification of constant mean curvature tori,
Ann. of Math. (2) 130 (1989), 407-451.
490
[23]
Nikolaos Kapouleas
R. Schoen, The existence of weak solutions with prescribed singular behavior for
a conformally invariant scalar equation, Comm. Pure Appi. Math. XLI (1988),
317-392.
[24] N. Smale, A bridge principle for minimal and constant mean curvature submanifolds
in RN, Invent. Math. 90 (1987), 505-549.
[25] C. H. Taubes, Self-dual Yang-Mills connections on non-self-dual ^.-manifolds, J.
Differential Geom. 17 (1982), 139-170.
[26] R. Walter, Explicit examples to the H-Problem of Heinz Hopf, Geom. Dedicata 23
(1987), 187-213.
[27] H. C. Wente, Counterexample to a conjecture of B. Hopf, Pacific J. Math. 121
(1986), 193-243.