


Document Library: Series 2 Abstracts

[2.4] 
Hoffman, D.
Some basic facts, old and new, about triply periodic embedded minimal surfaces
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[From the introduction]
In the last twenty years, triplyperiodic embedded minimal surfaces
have been of great interest and utility to researchers in chemistry,
cystallography and material science. The classical examples of Schwarz,
Neovious et al., dating from the 19th century were known, but seemed
to rest on the periphery of modern mathematical interest in minimal
surfaces. The monograph of Schoen, which may fairly be said to be the
source of modern interest in the sub ject, presented many new
examples, but this work was much better known to scientists than to
mathematicians. Except for the gyroid, few of Schoen's examples have
had very much impact, until quite recently, in the field of
mathematics proper. This has changed recently due in part to the work
of Karcher, FischerKoch, Wohlgemuth, Nitsche, and Ross.
 [2.5] 
Hoffman, D., Karcher, H. & Rosenberg, H.
Embedded minimal annuli in R^{3} bounded by a pair of straight lines
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[From the introduction]
The subject of this paper is embedded minimal annuli bounded by two
straight lines. The only known examples of such surfaces are given by
subdomains of the singly periodic Riemann examples, $R$. There is a
1parameter family of these surfaces. A fundamental domain of a
Riemann example consists of a minimal annulus bounded by two
straight lines, and a copy of that surface produced by Schwarz
reflection about one of the boundary lines. (See Figure 1. and the
analytic description of these surfaces in Section 2.)
 [2.8] 
Hoffman, D., Rosenberg, H. & Spruck, J.
Boundary value problems for surfaces of constant gauss curvature
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[From the introduction]
The compact smooth surfaces in R^{3} with constant positive Gauss
Curvature ($K$surfaces) form a natural class. A $K$surface without
boundary is itself the boundary of a convex body, so it must be
embedded. The surfaces of interest to us have nonempty boundary and
so are not necessarily embedded. A fundamental question is this: given
a collection $\gamma = \{C_1,\dots,C_n\}$ of Jordan curves in R^{3},
what are the $K$surfaces with boundary $\gamma$? For example, if
$\gamma$ is a single Jordan curve with no inflection points, does
$\gamma$ bound a $K$surface?
 [2.10] 
Spruck, J. & Yang, Y.
On multivortices in the electroweak theory II: existence of Bogomol'nyi solutions in R^{2}
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[From the introduction]
In Part I of this paper [11], we have proven the existence of
Abrikosov like periodic vortices for the bosonic sector models
proposed by Ambjorn and Olesen [4] of the full GlashowSalamWeinberg
electroweak theory, where the gauge group is $SU(2)\times U(1)$.
These solutions were found from a Bogomol'nyi system of first order
equations which take on a more complicated form than in the classical
abelian case due to the anti screening of the magnetic field. As a
result, this system further reduces to a semilinear elliptic system of
nonstandard type and we showed in Part I that the number of such
vortices is bounded above in terms of the relevant physical parameters
(although the locations may be prescribed arbitrarily). The goal of
the present paper is to study this Bogomol'nyi system for the self
dual electroweak interactions in the full space R^{2}. These solutions
are necessarily of infinite energy and thus the method of Part I
cannot be directly applied. Our main strategy then, is to combine the
method of weighted Sobolev spaces used by McOwen [8] in his study of
conformal deformation equations, with the crucial change of variables
introduced in Part I to reduce our elliptic system to a lower diagonal
form. As a result, we are able to show (Theorem 3.3) that for any
distribution of vortex locations there is a two parameter family of
gaugedistinct solutions.
 [2.11] 
Anderson, D. M., Bellare, J., Hoffman, J., Hoffman, D., Gunther, J. & Thomas, E. L.
Algorithms for the computer simulation of 2d projection from structures determined by dividing surfaces
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Transmission electron microscopy (TEM) data come in the form of two
dimensional projections of specimens with finite width. For visual
interpretation of such data it is valuable, and for quantitative
comparison with models essential, to have available a method for
generating simulated projections from model structures. We present
three computer algorithms that simulate TEM micrographs, from
structures determined by contrast between two materials separated by
complicated dividing surfaces. Three very general forms used to
represent dividing surfaces are treated: traditional coordinate
systems, finite element representations, and tesselations adapted to
complex line integration schemes. Because surface tension drives
dividing surfaces to minimize their area, particular attention is paid
to minimal surfaces, surfaces of nonzero constant mean curvature,
and to parallel surfaces that may form in specimens constrained to
small dimensions. Besides the model structure itself, the other inputs
are the orientation of the model structure, the location of (parallel)
upper and lower truncating planes representing the finite specimen
thickness or the form of the bounding surface for microdroplet
samples, and the direction and magnitude of a linear deformation
representing distortion due to the microtoming process. The values
of these registration parameters providing the optimal fit with
digitized TEM data are found by a relaxation method. Remarkable
matches are obtained between micrographs of block copolymer
morphologies and model structures determined by surfaces of constant
mean curvature.
 [2.12] 
Hoffman, D.
Computing minimal surfaces with and without conformal representation
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[From the introduction]
In this short presentation we will discuss two ways to compute minimal
surfaces. The first way takes advantage of the classical conformal
representation of a minimal surface using analytic data on the
surface. The second way involves solving the Plateau Problem for a
specific boundary, usually polygonal. Our interest is in complete and
properly embedded minimal surfaces. Since we will be discussing ways
to compute, there will have to be some finiteness imposed on the
problem. We will assume compactness, not of the surfaces themselves,
of course, but of the underlying conformal structure. According to
Osserman's Theorem, a complete minimal surface of finite total
curvature is conformally a compact Riemann surface that has been
punctured in a finite number of points, each point corresponding to an
end of the surface. In particular, such a surface has finite topology.
The converse is not true as the example of the helicoid shows; the
helicoid is singlyperiodic and not flat, so it has infinite total
curvature. Whether there are more examples is an open question. See
[?] for more details.
 [2.16] 
Korevaar, N. & Kusner, R.
The structure of constant mean curvature embeddings in Euclidean 3space
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[From the introduction]
In these notes we will sketch the known necessary conditions for
properly embedded surfaces $\Sigma\subset R^3$ which have nonzero
constant mean curvature and finite topology. Many examples of constant
mean curvature immersions and embeddings have been constructed in
recent years through the work of H. Wente [13], N. Kapouleas [4, 5],
H. Karcher [6], U. Pinkall and I. Sterling [12]. Immersions of tori
and cylinders can already have extremely complicated behavior [12,
13], so general immersed constant mean curvature surfaces must be
quite varied in structure. If \006 is required to be properly embedded,
it turns out that its structure can be characterized fairly
well. Not only is this structure interesting in its own right, but it
could also aid in the analysis of physical systems involving several
materials or phases, such as those described by the CahnHilliard
equation [2, 3], where it is believed that slowly evolving interfaces
may have timevarying, approximately constant mean curvature.
 [2.18] 
Korevaar, N., Kusner, R., Meeks III, W. H. & Solomon, B.
Constant mean curvature surfaces in hyperbolic space
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[From the introduction]
A basic problem in differential geometry is the global structure and
classification of complete properly embedded constant mean curvature
surfaces in threemanifolds of constant sectional curvature. The
models for these manifolds are threedimensional Euclidean space R^{3},
the three sphere S^{3}, and hyperbolic threespace H^{3}.
 [2.23] 
Whitaker, N.
Numerical solution of the HeleShaw equations using the point vortex method
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[From the introduction]
In ([19, 20]), Tryggvason and Aref use a boundary integral method and
the vortexincell method to evolve the interface between two fluids
in a HeleShaw cell. The method gives excellent results for
intermediate values of the nondimensional surface tension
parameter. The results are different from the predicted results of
McLean and Saffman for small surface tension. For large surface
tension, there are some numerical problems. In this paper, we
implement the method of Tryggvason and Aref but use the point vortex
method instead of the vortexincell method. A parametric spline is
used to represent the interface. The finger widths obtained agree well
with those predicted by McLean and Saffman. We conclude that the
method of Tryggvason and Aref can provide excellent results but that
the vortexincell method may not be the method of choice for extreme
values of the surface tension parameter. In a second method, we
represent the interface with a Fourier representation. In addition, an
alternative way of discretizing the boundary integral is used. Our
results are compared to the linearized theory and the results of
McLean and Saffman and are shown to be highly accurate.
 [2.24] 
Meeks III, W. H., Earp, R., Brito, F. & Rosenberg, H.
Structure theorems for constant mean curvature surfaces bounded by a convex planar curve
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[From the introduction]
A circle $C$ in R^{3} is the boundary of two spherical caps of constant
mean curvature H for any positive number $H$, which is at most the
radius of $C$. It is natural to ask whether spherical caps are the only
possible examples. Some examples of constant mean curvature immersed
tori by Wente [7] indicate that there are compact genusone immersed
constant mean curvature surfaces with boundary C that are approximated
by compact domains in Wente tori; however, this has not been
proved. Still one has the conjecture: Conjecture 1. A compact constant
mean curvature surface bounded by a circle is a spherical cap if
either of the following conditions hold: 1. The surface has genus 0
and is immersed; 2. The surface is embedded.
 [2.25] 
Meeks III, W. H. & White, B.
Minimal surfaces bounded by convex curves in parallel planes
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[From the introduction]
In 1956 M. Shiffman [17] proved several beautiful theorems concerning
the geometry of a minimal annulus $A$ whose boundary consists of two
closed convex curves in parallel planes $P_1, P_2$. The first theorem
stated that the intersection of $A$ with any plane $P$, between P_{1}
and P_{2}, is a convex Jordan curve. In particular it follows that $A$ is
embedded. He then used this convexity theorem to prove that every
symmetry of the boundary of $A$ extended to a symmetry of $A$. In the
case that @ $A$ consists of two circles Shiffman proved that $A$ was
foliated by circles in parallel planes. Earlier B. Riemann [15]
described, in terms of elliptic functions, all minimal annuli in R^{3}
that can be expressed as the union of circles in parallel planes (also
see [3] for a description of these surfaces as well as a computer
graphics image of one of them). Together these results yield a
classification of all minimal annuli with boundary consisting of
circles in parallel planes.
 [2.26] 
Meeks III, W. H. & Frohman, C.
The topological uniqueness of complete oneended minimal surfaces and Heegaard surfaces in R^{3}
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[From the introduction]
In this paper we shall prove two fundamental theorems on the
topological uniqueness of certain surfaces in R^{3}. The first of these
theorems, which will depend on the second theorem, shows that a
properly embedded minimal surface in R^{3} with one end is
unknotted. More precisely, Theorem 1.1 Two properly embedded
oneended minimal surfaces in R^{3} of the same genus are ambiently
isotopic.
 [2.28] 
Meeks III, W. H. & White, B.
The space of minimal annuli bounded by an extremal pair of planar curves
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[From the introduction]
In 1956 Shiffman [14] proved that every minimally immersed annulus in
R^{3} bounded by convex curves in parallel planes is embedded. He proved
this theorem by showing that the minimal annulus was foliated by
convex curves in parallel planes. We are able to prove a related
embeddedness theorem for extremal convex planar curves. Recall that a
subset of R^{3} is extremal if it is contained on the boundary of its
convex hull. We will call a pair of convex curves extremal if their
union is extremal.
 [2.29] 
Meeks III, W. H. & Frohman, C.
The ordering theorem for the ends of properly embedded minimal surfaces
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[From the introduction]
A fundamental problem in the classical theory of minimal surfaces is
to describe the asymptotic geometry of properly embedded minimal
surfaces in R^{3}. In the special case that the surface has finite
total curvature 1 its asymptotic behavior is well understood. For, in
this case, the surface is conformally diffeomorphic to a finitely
punctured closed Riemann surface and each end of the surface, one for
each puncture point, is asymptotic to a plane or an end of a catenoid
(see [19]). Thus the plane and the catenoid are the models for
describing the asymptotic behavior of these minimal surfaces. When the
properly embedded minimal surface has infinite total curvature, but
still finite topology, the question has been asked whether the surface
must be asymptotic to a helicoid.
 [2.30] 
Meeks III, W. H. & Rosenberg, H.
The geometry and conformal structure of properly embedded minimal surfaces of finite topology in R^{3}
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[From the introduction]
In this paper we study the conformal structure and the asymptotic
behavior of properly embedded minimal surfaces of finite topology in
R^{3}. One consequence of our study is that when such a surface has at
least two ends, then it has finite conformal type, i.e., it is
conformally diffeomorphic to a compact Riemann surface punctured in a
finite number of points.
 [2.32] 
Rosenberg, H.
Hypersurfaces of constant curvature in space forms
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[From the introduction]
In this paper we shall discuss hypersurfaces M of space forms of
constant curvature; where curvature means one of the symmetric
functions of curvature associated to the second fundamental form. The
values of the constant will be chosen so that the linearized equation
will be an elliptic equation on M . For example, for surfaces in R^{3}
the two possible curvatures are the mean curvature H and the Gaussian
curvature $K$. The linearized equation for $H$ is always elliptic and for
$K$ it is elliptic when the constant $K$ is positive. In hyperbolic
3space, the constant $K >  1$ yields an elliptic
equation. Hypersurfaces of constant scalar curvature S_{2} will be
elliptic when $S_2 > 0$.
 [2.34] 
Baginski, F. E.
The computation of oneparameter families of bifurcating elastic surfaces
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We consider the problem of constructing the middle surface of a
deformed elastic shell from its first and second fundamental forms,
$\hat{a}_{\alpha\beta}$ and $\hat{b}_{\alpha\beta}. The undeformed shell
is a spherical cap of radius
$R$ and thickness $h$ with an angular width $2\theta_0$ where
$0 < \theta_0 < \pi/2$. The cap is subjected to a constant uniform
load $\lambda$ and is
simply supported at its edge. We seek to compute the oneparameter
families of buckled states which branch from the unbuckled state of
the shell. This is accomplished in two steps. First, a finite element
method is used to solve the governing shell equations, a pair of
fourth{order nonlinear partial differential equations. A solution of
this system is a curvature potential $w$, a stress potential $f$, and the
load $\lambda$. Using LiapunovSchmidt reduction, it can be shown that
solutions possessing a variety of symmetries bifurcate from the
unbuckled state of the shell. In the work that is presented here, we
will numerically continue these local branches. We parametrize
solution branches in terms of a pseudoarclength parameter $\rho
(i.e., $(\lambda,f,w)=(\lambda(\rho),f_\rho,w_\rho)$), enabling us
to track them around turning points. The second step in our solution
process is to solve numerically for the parametrization $\hat{X}_\rho$
corresponding to the middle surface of the buckled shell $\hat{X}_\rho$. We
do so by integrating the partial differential equations of $\hat{S}_\rho$.
The coefficients in these differential equations involve the first and
second fundamental forms of the deformed shell $\hat{S}_\rho$ which can be
computed from $(\lambda(\rho),f_\rho,w_\rho)$. A number of
bifurcation diagrams corresponding to the first three branch points of
a spherical cap of size $\theta_0=12.85\degree$ are presented. For this
example, a secondary bifurcation point was found connecting two
distinct nonaxisymmetric solution branches. Computer graphics are used
to display images of various buckled surfaces which branch from the
unbuckled state of the shell.
 [2.39] 
Romon, P.
A rigidity theorem for Riemann's minimal surfaces
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We describe first the analytic structure of Riemann's examples of
singlyperiodic minimal surfaces; we also characterize them as
extensions of minimal annuli bounded by parallel straight lines
between parallel planes. We then prove their uniqueness as solutions
of the perturbed problem of a punctured annulus, and we present
standard methods for determining finite total curvature periodic
minimal surfaces and solving the period problems.
 [2.40] 
Kusner, R.
The number of faces of a minimal foam
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[From the introduction]
A compound bubble is a partition of a Riemannian 3manifold M into
domains whose boundaries are smooth constant mean curvature surfaces,
meeting 3 to a smooth edge or 6 to an isolated vertex, at equal
angles. Observe that the equal angle condition means that at any point
on the support $\Sigma$ (that is, the union of the boundary surfaces) of
the compound bubble the tangent cone is either a plane, the product of
a line with an equiangular "Y" or the central cone over the
1skeleton of a regular tetrahedron.


© GANG 2001
