407 research outputs found
A characterisation of the Hoffman-Wohlgemuth surfaces in terms of their symmetries
For an embedded singly periodic minimal surface M with genus bigger than or
equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence
with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof
of this fact, along which they come out many valuable clues for the
understanding of these surfaces
An embedded genus-one helicoid
There exists a properly embedded minimal surface of genus one with one end.
The end is asymptotic to the end of the helicoid. This genus one helicoid is
constructed as the limit of a continuous one-parameter family of screw-motion
invariant minimal surfaces--also asymptotic to the helicoid--that have genus
equal to one in the quotient.Comment: 115 pages, many figures Minor exposiitonal changes and added
reference
Helicoids and vortices
We point out an interesting connection between fluid dynamics and minimal
surface theory: When gluing helicoids into a minimal surface, the limit
positions of the helicoids correspond to a "vortex crystal", an equilibrium of
point vortices in 2D fluid that move together as a rigid body. While vortex
crystals have been studied for almost 150 years, the gluing construction of
minimal surfaces is relatively new. As a consequence of the connection, we
obtain many new minimal surfaces and some new vortex crystals by simply
comparing notes.Comment: 16 pages, 5 figures. Good news: Confirmed Adler--Moser translating
examples; added. Bad news: None of the non-trivial stationary examples gives
actual example; remove
Cubic Polyhedra
A cubic polyhedron is a polyhedral surface whose edges are exactly all the
edges of the cubic lattice. Every such polyhedron is a discrete minimal
surface, and it appears that many (but not all) of them can be relaxed to
smooth minimal surfaces (under an appropriate smoothing flow, keeping their
symmetries). Here we give a complete classification of the cubic polyhedra.
Among these are five new infinite uniform polyhedra and an uncountable
collection of new infinite semi-regular polyhedra. We also consider the
somewhat larger class of all discrete minimal surfaces in the cubic lattice.Comment: 18 pages, many figure
Helicoidal minimal surfaces of prescribed genus, I
For every genus g, we prove that S^2 x R contains complete, properly
embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids
of any prescribed pitch. We also show that as the radius of the S^2 tends to
infinity, these examples converge smoothly to complete, properly embedded
minimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In a
companion paper, we prove that helicoidal surfaces in R^3 of every prescribed
genus occur as such limits of examples in S^2 x R.Comment: 53 pages, 5 figure
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