13 research outputs found
Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I
We construct closed embedded minimal surfaces in the round three-sphere,
resembling two parallel copies of the equatorial two-sphere, joined by small
catenoidal bridges symmetrically arranged either along two parallel circles of
the equator, or along the equatorial circle and the poles. To carry out these
constructions we refine and reorganize the doubling methodology in ways which
we expect to apply also to further constructions. In particular we introduce
what we call linearized doubling, which is an intermediate step where singular
solutions to the linearized equation are constructed subject to appropriate
linear and nonlinear conditions. Linearized doubling provides a systematic
approach for dealing with the obstructions involved and also understanding in
detail the regions further away from the catenoidal bridges.Comment: Final version to appear in JDG. 48 pages, no figure
The Lawson surfaces are determined by their symmetries and topology
We prove that a closed embedded minimal surface in the round three-sphere
which satisfies the symmetries of a Lawson surface and has the same genus is
congruent to the Lawson surface.Comment: 15 pages, no figures, with minor improvement
Mean curvature self-shrinkers of high genus: Non-compact examples
We give the first rigorous construction of complete, embedded self-shrinking
hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The
surfaces exist for any sufficiently large prescribed genus , and are
non-compact with one end. Each has symmetries and comes from
desingularizing the intersection of the plane and sphere through a great
circle, a configuration with very high symmetry. Each is at infinity asymptotic
to the cone in over a -periodic graph on an equator
of the unit sphere , with the shape of a
periodically "wobbling sheet". This is a dramatic instability phenomenon, with
changes of asymptotics that break much more symmetry than seen in minimal
surface constructions. The core of the proof is a detailed understanding of the
linearized problem in a setting with severely unbounded geometry, leading to
special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the
gradient terms. This involves identifying new, adequate weighted H\"older
spaces of asymptotically conical functions in which the operators invert, via a
Liouville-type result with precise asymptotics.Comment: 41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew.
Mat
Special Lagrangian cones with higher genus links
For every odd natural number g=2d+1 we prove the existence of a countably
infinite family of special Lagrangian cones in C^3 over a closed Riemann
surface of genus g, using a geometric PDE gluing method.Comment: 48 page