


Document Library: Series 3 Abstracts

[3.2] 
Hoffman, D., Wei, F. & Karcher, H.
Adding handles to the helicoid
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[Excerpt]
We have constructed two minimal surfaces of theoretical interest. The
first is a complete, embedded, singlyperiodic minimal surface (SPEMS)
that is asymptotic to the helicoid, has infinite genus, and whose
quotient by translations has genus one. The quotient of the helicoid
by translations has genus zero and the helicoid itself is
simplyconnected.
 [3.6] 
Hoffman, D. & Karcher, H.
Complete embedded minimal surfaces of finite total curvature
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[From the introduction]
We will survey what is known about minimal surfaces $S\subset\R^3$,
which are complete, embedded, and have finite total curvature: \[\].
The only classically known examples of such surfaces were the
plane and the catenoid. The discovery by Costa [14, 15], early in the
last decade, of a new example that proved to be embedded sparked a
great deal of research in this area. Many new examples have been
found, even families of them, as will be described below. The central
question has been transformed from whether or not there are any
examples except surfaces of rotation to one of understanding the
structure of the space of examples.
 [3.10] 
Meeks III, W. H. & Rosenberg, H.
The geometry of periodic minimal surfaces
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[From the introduction]
In this paper we shall demonstrate a surprising relationship between
the topology of a properly embedded periodic minimal surface in R^{3}
and its global geometry. We shall call a minimal surface periodic if
it is connected and invariant under a group $G$ of isometries that acts
freely on R^{3}. We will analyze these surfaces by studying their
quotient in $R^3/G$. We have already carried out this study for
doublyperiodic minimal surfaces [16].
 [3.12] 
Kim, D. & Kusner, R.
Torus knots extremizing the conformal energy
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[From the introduction]
Recently Freedman, He and Wang [FHW], following work of O'Hara [O],
introduced an energy $E(\Gamma)$ for a simple closed curve
$\Gamma\subset R^3$. The functional $E$ is continuous on each isotopy
class of curves, and tends to infinity as $\Gamma$ nears
selfintersection. Moreover, $E$ is "proper" on the set of all isotopy
classes, in the sense that there are only finitely many knot types below
a given energy level.
 [3.13] 
Hoffman, D. & Rossman, W.
Limit surfaces of Riemann examples
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[From the introduction]
The only connected minimal surfaces foliated by circles and lines are
domains on one of the following surfaces: the helicoid, the catenoid,
the plane, and the examples of Riemann ([Ri] p32933, [En] p4036,
[Ni] p856). All these surfaces are complete and embedded.
Topologically they are planar domains: the helicoid is
simplyconnected, the catenoid is an annulus (conformally a
twicepunctured sphere), and each Riemann example (see Figure 1) is
conformal to the plane minus the points
$\{(n,0),(\frac{1}{n},0)n\in Z\} [HKR].
In this section we will show that the plane, helicoid, and
catenoid arise naturally as limits of wellchosen and properly
normalized sequences of Riemann examples. The local behavior of
domains on Riemann examples accounts for the existence of these
limits, thus allowing a change of topology in the limit surfaces.
 [3.14] 
Bolton, J., Pedit, F. & Woodward, L.
Minimal surfaces and the affine Toda field model
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[From the introduction]
Minimal immersions or, more generally, harmonic maps of a Riemann
surface $S$ into $S^n$; $CP^n$ and other Riemannian symmetric spaces have
been intensively studied over the past twenty five years. The subject
has been given considerable impetus by the interest of mathematical
physicists in nonlinear $\sigma$models, which are harmonic maps of
S^{2} into $CP^n$, and in related problems which may be handled using
twistor theory and methods of complex geometry.
 [3.15] 
Burstall, F. E. & Pedit, F.
Harmonic maps via AdlerKostantSymes theory
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[From the introduction]
Over the past few years significant progress has been made in the
understanding of various completely integrable nonlinear PDE (soliton
equations) and their relationship to classical problems in
differential geometry. It has been shown in a series of recent papers
[41, 31, 20, 24, 14, 7, 11] that constant mean and Gauss curvature
surfaces, Willmore surfaces, minimal surfaces in spheres and projective
spaces and generally harmonic maps from a Riemann surface M
into various homogeneous spaces may be described as solutions to
various soliton equations. Moreover, these solutions are algebraic in
the sense that they are obtained by integrating ordinary differential
equations of Lax type which linearise on the Jacobian of an
appropriate algebraic curve.
 [3.16] 
Hoffman, D., Karcher, H. & Wei, F.
The genusone helicoid and the minimal surfaces that led to its discovery
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[From the introduction]
Around 1780, soon after Lagrange derived the minimal surface equation,
Meusnier found the first nonplanar solutions: Euler's catenoid and
the helicoid. More than 200 years later, in 1980, the catenoid was
still the only known finite total curvature embedded minimal surface
and the helicoid was the only known infinite total curvature embedded
minimal surface of finite topology. First, the finite total curvature
situation changed. ChenGackstatter [CG] found an immersed torus with
one end and  soon after that  Costa [C] found an embedded torus with
three ends. We now know that for every genus $\ge$ 1, there exists a
oneparameter family of embedded finite total curvature minimal
surfaces with three ends ([HM]), and we have some more examples  less
well understood  with four and five ends. ([CHM,BW,W1,W2,HK]).
In this paper we describe the construction of a minimally embedded
torus with one end and infinite total curvature ([HKW]). The
(elliptic) Gauss map of this surface has an essential singularity at
the puncture and this makes the surface fundamentally different from
all previous embedded examples. The other infinite total curvature and
finite topology embedded minimal surface, the helicoid, can be
described with the exponential map as Gauss map, i.e. with an
essential singularity. The new surface is substantially more
complicated than the helicoid because it does not cover a finite total
curvature surface in a flat spaceform (namely in R^{2}\times S^1$ in the
case of the helicoid). Therefore it cannot be described on a quotient
surface in terms of a meromorphic Gauss map. (In the case of the
helicoid, $g(z)=z$ on $C\setminus\{0\}$.)
 [3.17] 
Baginski, F. E. & Whitaker, N.
Numerical Solutions of Boundary Value Problems for kSurfaces in R^{3}
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A $K$surface is a surface whose Gauss curvature $K$ is equal to a
positive constant. In this paper, we will consider $K$surfaces that are
defined by a nonlinear boundary value problem. In this setting,
existence follows from some recent results on nonlinear secondorder
elliptic partial differential equations. The analytical techniques
used to establish these results motivate effective numerical methods
for computing $K$surfaces. In theory, the solvability of the boundary
value problem reduces to the existence of a subsolution. In an
analogous way, if an approximate numerical subsolution can be
determined, then the corresponding $K$surface can be computed. We will
consider two boundary value problems. In the first problem, the
$K$surface is a graph over a plane. In the second problem, the
$K$surface is a radial graph over a sphere. From certain geometrical
considerations, it follows that there is a maximum allowable Gauss
curvature $K_{max} for these problems. The principal results in this
paper are numerical estimates of $K_{max} for a variety of geometries and
boundary data. Using a continuation method, we determine numerically
the unique oneparameter family of $K$surfaces that exist for
$K$ \in (0, $K_{max} ).
We can compare our numerical estimates for $K_{max} to the true
value when the $K$surface is a subset of a hyperbolic spherical
surface of revolution. In this case, we find that our numerical
estimates for $K_{max} are in close agreement with the expected values.
 [3.20] 
Callahan, M., Hoffman, D. & Karcher, H.
A family of singlyperiodic minimal surfaces invariant under a screw motion
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[From the introduction]
In [2], two of the authors and W. Meeks found examples of
translationinvariant, embedded minimal surfaces with an infinite
number of topological ends. For each $k>0$, a surface $M_k$ was
constructed, which was invariant with respect to a translation
parallel to the x_{3}axis, and under a rotation group of order $k+1$
around the x_{3}axis. The method of construction was generalized in
[3] to obtain the first known examples of embedded, singlyperiodic
minimal surfaces with an infinite number of topological ends,
invariant under screwmotions (with a nontrivial rotational
component). For each integer $k>0$ and angle $\theta$, with
$\theta < \frac{\pi}{k+1}$, there exists an embedded surface
$M_{k,\theta} whose
orientationpreserving symmetry group contains a rotation of order
$k+1$ around the x_{3} axis and a screw motion  a unit translation in
the x_{3}direction, followed by a $2\theta$ rotation around it. See
Figure 0. Although the surfaces $M_{k,\theta}$ were conceived1 as smooth
deformations of the singlyperiodic examples $M_k$, the proof in [3]
did not construct these deformations. In fact, it left open the
following questions: 1. Is the family $M_{k,\theta} smooth in $\theta$?
If so, are they deformations of $M_k$? 2. Are the surfaces $M_{k\theta}
unique? 3. Is there a surface $M_{k,\pi/(k+1)$? To be more precise: the
symmetry groups of the $M_{k,\theta}$ have a single limit as
$\theta\leftarrow\pi/(k+1)$ and $\theta\leftarrow \pi/(k+1)$.
This group contains a translation but is different from the symmetries of
$M_{k,\theta}. The
question becomes, is there a surface with this symmetry group and
obvious properties generalizing the $M_{k,\theta}$?
 [3.23] 
Kusner, R. & Sullivan, J.
Möbius energies for knots and links, surfaces and submanifolds
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There has been recent interest in knot energies, especially
those which are invariant under Möbius transformations of
space. We describe computer experiments with such energies, and
discuss ways of extending these to energies for higherdimensional
submanifolds. The appendix gives a table of computed energyminimizing
knots and links through eight crossings.
 [3.24] 
Pedit, F. & Wu, H.
Discretizing constant curvature surfaces via loop group factorizations: the discrete sine and sineGordon equations
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[From the introduction]
In recent years mathematical physicists have been studying discrete
(in space and time) analogs of integrable nonlinear field models
motivated by questions arising in statistical mechanics (spin models)
and quantum field theory [10]. Perhaps it is necessary to explain what
we mean by integrable . We call a nonlinear field equation
integrable if it arises as the flatness (zerocurvature) condition of
a connection with values in a loop Lie algebra. A standard example for
this is the nonlinear \033model, i.e., the harmonic map equation for
maps of a surface into a symmetric space [14, 19, 18, 7].
 [3.25] 
Dorfmeister, J., Pedit, F. & Wu, H.
Weierstrass type representation of harmonic maps into symmetric spaces
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[From the introduction]
Over the past five years substantial progress has been made in the
understanding of harmonic maps $f:M\rightarrow G/K$ of a compact
Riemann surface M into a compact symmetric space $G=K$ [29, 30, 12,
14, 17, 22, 5, 7, 10, 31]. When $M$ is the Riemann sphere all such
maps may b e obtained from holomorphic curves into some associated
twistor space [11, 29, 14]. In contrast, if $M$ is a 2torus, then
every harmonic map $\phi:M\rightarrow G/K$ (satisfying specific nondegeneracy
assumptions) may be obtained from a solution to a certain family of
completely integrable finite dimensional systems of Hamiltonian ODE
in Lax form on a loop algebra [5, 7, 17]. Such harmonic maps are
called harmonic maps of finite type and, e.g., all harmonic 2tori in
$S^n$, $CP^n$ are accounted for in this way [4]. So far there is no
comparable systematic theory when $M$ has higher genus. Nevertheless
there are examples: the harmonic maps $f:M\rightarrow S^2$ arising as Gauss
maps of Kapouleas' [18, 19] constant mean curvature surfaces of genus
$g\ge 2$ and the minimal (and hence harmonic) maps $f:M \rightarrow S^3$
given by Lawson's [20] minimal surfaces of genus $g\ge 2$.
 [3.27] 
Kusner, R. & Schmitt, N.
The spinor representation of minimal surfaces (superseded)
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[From the introduction]
In this paper we investigate the interplay between spin structures on
a Riemann surface $M$ and immersions of $M$ into threespace. Here, a
spin structure is a complex line bundle S over $M$ such that $S\otimes S$
is the holomorphic (co)tangent bundle $T(M)$ of $M$. Thus we may view a
section of a S as a "square root" of a holomorphic 1form on M
. Using this notion of spin structure, in the first part of this paper
we develop the notion of the spinor representation of a surface in
space , based on an observation of Dennis Sullivan [27]. The classical
Weierstrass representation is \[\] where $g$ and $\eta$ are respectively
a meromorphic
function and oneform on the underlying compact Riemann surface. The
spinor representation (Theorem 5) is \[\] where s_{1} and s_{2} are
meromorphic sections of a spin structure S . Either representation
gives a (weakly) conformal harmonic map $M\rightarrow R^3$ , which
therefore parametrizes a (branched) minimal surface.


© GANG 2001
