72 research outputs found
Spacelike CMC 1 surfaces with elliptic ends in de Sitter 3-Space
We show that an Osserman-type inequality holds for spacelike surfaces of
constant mean curvature (CMC) 1 with singularities and with elliptic ends in de
Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if
the monodromy representation at the end is diagonalizable with eigenvalues in
the unit circle. We also give a necessary and sufficient condition for equality
in the inequality to hold, and in the process of doing this we derive a
condition for determining when elliptic ends are embedded.Comment: 23 pages, 6 figures. v2: Section 3 added to give a criterion for
embeddedness of elliptic ends. Corollary 5.8 added to show the existence of
uncountably many CMC 1 faces from CMC 1 immersions in [MU]. New references
added. v3: revision according to the referee's suggestion
Triply periodic zero mean curvature surfaces in Lorentz-Minkowski 3-space
We construct triply periodic zero mean curvature surfaces of mixed type in
the Lorentz-Minkowski 3-space, with the same topology as the triply periodic
minimal surfaces in the Euclidean 3-space, called Schwarz rPD surfaces.Comment: 16 pages, 9 figure
Minimal surfaces with two ends which have the least total absolute curvature
In this paper, we consider complete non-catenoidal minimal surfaces of finite
total curvature with two ends. A family of such minimal surfaces with least
total absolute curvature is given. Moreover, we obtain a uniqueness theorem for
this family from its symmetries.Comment: 37 pages, 17 figure
Triply Periodic Minimal Surfaces Bounded by Vertical Symmetry Planes
We give a uniform and elementary treatment of many classical and new triply
periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel
formula for periodic polygons in the plane. Our surfaces share the property
that vertical symmetry planes cut them into simply connected pieces.Comment: 30 pages, 56 figure
Loop Group Methods for Constant Mean Curvature Surfaces
This is an elementary introduction to a method for studying harmonic maps
into symmetric spaces, and in particular for studying constant mean curvature
(CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit and H. Wu. There
already exist a number of other introductions to this method, but all of them
require a higher degree of mathematical sophistication from the reader than is
needed here. The authors' goal was to create an exposition that would be
readily accessible to a beginning graduate student, and even to a highly
motivated undergraduate student. Constant mean curvature surfaces in Euclidean
3-space, and also spherical 3-space and hyperbolic 3-space, are described,
along with the Lax pair equations that determine their frames. The simplest
examples, including Delaunay surfaces and Smyth surfaces, are described in
detail.Comment: This is an introductory exposition on constructing constant mean
curvature surfaces by techniques of integrable systems. A version with higher
quality graphics exists at the home page of the Rokko Lectures in Mathematics
series. Version 2: six minor errors repaired, and one figure repaire
Singularities of Maximal Surfaces
We show that the singularities of spacelike maximal surfaces in
Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails
and cuspidal cross caps. The same result holds for spacelike mean curvature one
surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion
for a given singular point on a surface to be a cuspidal cross cap.Comment: 20 pages, 2 figure
Higher genus mean curvature 1 catenoids in hyperbolic and de Sitter 3-spaces
We show existence of constant mean curvature 1 surfaces in both hyperbolic
3-space and de Sitter 3-space with two complete embedded ends and any positive
genus up to genus twenty. We also find another such family of surfaces in de
Sitter 3-space, but with a different non-embedded end behavior.Comment: 11 pages, 5 figure
Spacelike mean curvature one surfaces in de Sitter 3-space
The first author studied spacelike constant mean curvature one (CMC-1)
surfaces in de Sitter 3-space when the surfaces have no singularities except
within some compact subset and are of finite total curvature on the complement
of this compact subset. However, there are many CMC-1 surfaces whose singular
sets are not compact. In fact, such examples have already appeared in the
construction of trinoids given by Lee and the last author via hypergeometric
functions. In this paper, we improve the Osserman-type inequality given by the
first author. Moreover, we shall develop a fundamental framework that allows
the singular set to be non-compact, and then will use it to investigate the
global behavior of CMC-1 surfaces.Comment: 32 pages, 3 figure
Embedded triply periodic zero mean curvature surfaces of mixed type in Lorentz-Minkowski 3-space
We construct embedded triply periodic zero mean curvature surfaces of mixed
type in the Lorentz-Minkowski 3-space with the same topology as the Schwarz D
surface in the Euclidean 3-space.Comment: 17 pages, 14 figure
Zero mean curvature entire graphs of mixed type in Lorentz-Minkowski 3-space
It is classically known that the only zero mean curvature entire graphs in
the Euclidean 3-space are planes, by Bernstein's theorem. A surface in
Lorentz-Minkowski 3-space is called of mixed type if it
changes causal type from space-like to time-like. In ,
Osamu Kobayashi found two zero mean curvature entire graphs of mixed type that
are not planes. As far as the authors know, these two examples were the only
known examples of entire zero mean curvature graphs of mixed type without
singularities. In this paper, we construct several families of real analytic
zero mean curvature entire graphs of mixed type in Lorentz-Minkowski -space.
The entire graphs mentioned above lie in one of these classes.Comment: 31 pages, 5 figure
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