New Constant Mean Curvature TrinoidsAugust 23, 1999 Abstract I construct two families of constant mean curvature genus-zero surfaces with three ends via the DPW construction. One of these families is known and has three unduloid ends; the other is a new family with two unduloid and one nodoid end. The classification of the almost-embedded CMC genus-zero surfaces with three ends -- the trinoids with unduloid ends, or triunduloids -- was announced by Große-Brauckmann, Kusner, and Sullivan [2]. In this note, I construct these examples explicitly (result 1) using a method due to Dorfmeister, Pedit and Wu [1], and present a new family of CMC trinoids (result 2) which enjoy two unduloid and one nodoid end. Taking H>0 to be a fixed mean curvature, the space of triunduloids may be described as follows.
Result 1
Let
be real numbers in the interval
 satisfying
Then there exist two potentials which, with appropriate initial conditions, produce trinoids with
as neck radii,
except when any of the above four inequalities is an
equality; in this case there is exactly one such trinoid.
We extend this result to trinoids with two unduloid and and one nodoid end
(that is, ends asymptotic to Delaunay unduloids and nodoids respectively).
Result 2
Let
be real numbers in the intervals
,
,
respectively
satisfying
Suppose further that the end weights 
corresponding
to the neck radii
satisfy
 .
Then there exist two potentials which, with appropriate
initial conditions, produce trinoids with
as neck radii,
except when any of the above three inequalities
for
is an equality;
in this case there is one such trinoid.These surfaces were produced with two tools I developed at the Center for Geometry, Analysis, Numerics and Graphics (GANG). The algorithm for the main software, DPWLab, which computes CMC surfaces from DPW potentials, is discussed in [1]. The second tool, killerB, discussed in more detail below, uses an unsophisticated but very stable algorithm to compute the initial condition for which a surface simultaneously closes at its ends.
Neck Radii and End Weights. The minimum and maximum radii 
of a Delaunay unduloid
satisfy
0<a<1/(2H)<b<1/H, with constant sum a+b=1/H.
It is natural to extend this notion of neck and bulge radii
to the round cylinder
(
a=b=1/(2H)), the round sphere (
 ), and
to the nodoid, where a<0<1/H<b.
In all cases, the sum is a+b=1/H.
The end weight, defined as w=ab, is at most 1/(4H) (for the round cylinder) and like the neck radius, is positive for unduloids, zero for the round sphere, and negative for nodoids. The Family of Trinoid Potentials. The family of DPW potentials generating the trinoids of results 1 and 2 are meromorphic on the thrice-punctured Riemann sphere and are asymptotic to the DPW potential for a Delaunay surface at each puncture. For 
and
 I denote this potential by
The Delaunay surface produced by this potential is closed if and only if 
satisfy
Fix distinct points  .
The space of trinoid potentials is parameterized by
 as follows. Fix 
and let
s=a1+a2+a3.
Let
satisfy
 .
Here, 
denotes the cross ratio
Then the trinoid potential  is the unique form with divisor
-[e1]-[e1]-[e3]-[d]and residues
 
is the potential for a (closed) constant mean curvature immersion
 under the DPW procedure for initial condition found numerically.
The resulting trinoid will be denoted  .
The Parameter Space Conditions. Of the two conditions defining U, 
(3) insures that at each of its poles  ,
the Delaunay potential (1)
to which
is asymptotic satisfies the
closing condition (2).
The remaining condition (4) defining Uarises from a balancing formula [3]
according to which the sum of the forces exerted by each end
is zero. It follows that the end weights must satisfy the
inequality
Each trinoid
On the other hand, there are
two trinoids for each choice of neck radii unless v=0.
The two values End Types. The family of trinoids parametrized by
must have either
(a) three unduloid ends, or (b) two unduloid and one nodoid end.
Trinoids with more than one nodoid end cannot arise in this
family.
To see this, note that
the ends
are unduloidal or nodoidal according as
their weights
4a1(a2+a3),
4a2(a1+a3),
4a3(a1+a2) are
positive or negative respectively.
Because
and
give the same trinoid, as do
permutations of
,
without loss of generality there are two cases:
(a) a1>0, a2>0, a3>0, and (b) a1>0, a2>0, a3<0.
Case (b) yields a trinoid three unduloid ends.
In case (b),
a3(a1+a2)<0, making e3 unduloidal.
The condition
a1 a2 + a1 a3 + a2 a3 > 0means that
a1(a2+a3) > -a2 a3 and
a2(a1+a3) > -a1 a3.
But
-a2 a3 > 0, so
a1(a2+a3) > -a2 a3 > 0, and likewise
-a1 a3 > 0, so
a2(a1+a3) > -a1 a3 > 0.
Hence each of the remaining two ends are unduloidal.
Asymptotics. At each of its poles  the z-leading-order term of is the DPW potential for a closed Delaunay surface Di,
and the trinoid
converges to Di near ei.
Near e1, Cone Points. The point ,
while a simple pole of ,
is
a regular point for the resulting immersion.
Writing
the cross-ratio condition for d guarantees that  ,
so the Hopf form has no pole
at d.
Symmetry. If 
is the anticonformal diffeomorphism
fixing
,
then
for a
reflection R of
 .
Hence each trinoid
has
a plane of reflective symmetry.
Umbilics. Each trinoid
has two umbilic points,
whose preimages
are the zeros of the coefficient
of
in the lower-right entry of .
For triunduloids 
are interchanged by the involution
and hence by the reflection
of
.
For trinoids with one nodoid end,
are individually
fixed by
and lie in the plane of reflective symmetry of
the trinoid.
Period Closing. As the DPW potentials for these surfaces became clear, I implemented period-closing software killerB to solve the holonomy problem numerically and verify that the potentials indeed produce trinoids. Since the desired initial condition is not a constant but a loop group element, the period closer's search space is an infinite dimensional space of Fourier series, so for the purposes of computation it is necessary to restrict to a finite dimensional subspace by truncating the series. The period-closing software uses a brute-force algorithm. Starting with an arbitrary initial guess, it cycles through the dimensions of the search space (the loop group coefficients), varying one parameter at a time while holding the others fixed. In this way it minimizes, one dimension at a time, an error measure of the holonomy by quadratic interpolation. By stair-stepping through the parameters, the algorithm slowly converges to a stable relative minimum which in practice is the global minimum. It is interesting to note that the periods of ends with small neck radii converge much more quickly than do those near the maximal limit. As noted above, due to the special structure of the potential, at one end a linear space of initial conditions can be found for which the trinoid closes. Within this space, the period-closing software must be used to close a second end. Because the holonomy around the third end coincides with the composition of holonomies around the other two, the closing condition on the third end is vacuous. Acknowledgments. I thank Ian McIntosh, Martin Kilian, and Rob Kusner. This work was partially supported by NSF grants DMS-9626804 and DMS-9704949. Bibliography
Center for Geometry, Analysis, Numerics and Graphics |
for each permutation 
of
produce the same trinoid with
ends permuted, and (b)
trivially produce the same surface.
give
two triunduloids with identical neck radii but opposite phase:
one has a bulge in the middle, and the other a neck. This
phenomenon is illustrated in figures
1 and 5.
