5,203 research outputs found
Triunduloids: Embedded constant mean curvature surfaces with three ends and genus zero
In 1841, Delaunay constructed the embedded surfaces of revolution with
constant mean curvature (CMC); these unduloids have genus zero and are now
known to be the only embedded CMC surfaces with two ends and finite genus.
Here, we construct the complete family of embedded CMC surfaces with three ends
and genus zero; they are classified using their asymptotic necksizes. We work
in a class slightly more general than embedded surfaces, namely immersed
surfaces which bound an immersed three-manifold, as introduced by Alexandrov.Comment: LaTeX, 22 pages, 2 figures (8 ps files); full version of our
announcement math.DG/9903101; final version (minor revisions) to appear in
Crelle's J. reine angew. Mat
Surfaces of constant curvature in R^3 with isolated singularities
We prove that finite area isolated singularities of surfaces with constant
positive curvature in R^3 are removable singularities, branch points or
immersed conical singularities. We describe the space of immersed conical
singularities of such surfaces in terms of the class of real analytic closed
locally convex curves in the 2-sphere with admissible cusp singularities,
characterizing when the singularity is actually embedded. In the global
setting, we describe the space of peaked spheres in R^3, i.e. compact convex
surfaces of constant positive curvature with a finite number of singularities,
and give applications to harmonic maps and constant mean curvature surfaces.Comment: 28 page
On the nondegeneracy of constant mean curvature surfaces
We prove that many complete, noncompact, constant mean curvature (CMC)
surfaces are nondegenerate; that is, the Jacobi operator
has no kernel. In fact, if has genus zero
and is contained in a half-space, then we find an explicit upper
bound for the dimension of the jernel in terms of the number of
non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields
which linearizes the conjugate cousin construction. Consequences include
partial regularity for CMC moduli space, a larger class of CMC surfaces to use
in gluing constructions, and a surprising characterization of CMC surfaces via
spinning spheres.Comment: v2: substantial revisions, to appear in Geom. Funct. Anal.; three
figure
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