For along time
the sphere has been the only known compact immersed surface of
constant mean curvature (cmc). By the result of Alexandrov
there is no other embedded compact cmc surface, and by a
theorem of Hopf the sphere is the only immersed cmc surface
with genus 0. Wente's surprising discovery of cmc tori in 1986
became the starting point of an intensive study of cmc
surfaces. We construct numerically new compact cmc
surfaces of genus greater than 2. Our surfaces are highly symmetric
and more explicit than examples constructed with Kapouleas' method.
The construction starts with a balanced graph (the topological retract
of a surface), where we place at each vertex a sphere and join them
along edges by a fundamental segment of a Delaunay problem in S3 whose solution conjugates to a
cmc patch in R3 with
the same symmetry as the graph and which produces the complete surface
via symmetry operations. Our examples come in three dihedrally
symmetric families, with genus ranging from 3 to 5, 7 to 10, and 3 to
9, respectively, there are further surfaces with the symmetry of the
Platonic polyhedra and genera 6, 12, and 30. We use the algorithm of
Oberknapp and Polthier,
which defines a discrete version of Lawson's conjugate surface method.
The fundamental domain for the spheroidal surface of genus 8. A polygon of five
great circle arcs in
S2cS3,
or, in the chosen stereographic projection, close to a plane. There are two
heliocoidal regions, one connecting the triangle to the two-gon, the other
in a neighbourhood of gamma2.
The isometric conjugate cmc patch. Its five boundary arcs
are contained in three different planes that meet pairwise in lines
shown. Thirtytwo reflected copies generate the compact surface
depicted in this Figure. The almost-planar regions of the previous
figure give spherical regions whilst the helicoidal regions result in necks.
These can be nodoidal (at gamma2),
or unduloidal (in between gamma3
and gamma5)
depending on the sense of rotation of the helicoids.
A surface of genus 6 with the symmetry of a tetrahedron. One
bubble is slightly moved. Six unduloidal necks join the four
outer bubbles pairwise. The central bubble looks lika a shell
punctured in four points to connect it nodoidally
to each outer bubble.
Modification of the Wente torus by attaching six Delaunay unduloid ends.
The result is a genus one constant mean curvature surface (left). The six
ends have parallel axes. The images on the right explain the building blocks
used in the construction: two bubbles of a rotationally symmetric Delaunay unduloid,
and two views of a Wente torus - two quadrilateral fundamental domains are shown as grid.
Compact Constant Mean Curvature Surface
