


Document Library: Series 5 Abstracts

[5.1] 
GrosseBrauckmann, K., Kusner, R. & Sullivan, J.
Constant mean curvature surfaces with three ends
postscript
abstract
We announce the classification of complete, almost embedded surfaces
of constant mean curvature, with three ends and genus zero: they are
classified by triples of points on the sphere whose distances are the
asymptotic necksizes of the three ends.
 [5.2] 
HertrichJeromin, U., McIntosh, I., Norman, P. & Pedit, F.
Periodic discrete conformal maps
postscript
abstract
[From the introduction]
Recently there has been much interest in the theory of discrete
surfaces in 3 space and its connection with the discretization of
soliton equations (see e.g. [4, 11] and references therein). In this
article we study a discrete geometry which is the simplest example for
both theories. Following [1, 3] we will define a discrete conformal
map (DCM) to be a map $z:Z^2\rightarrow P^1$ with the property that
the crossratio of each fundamental quadrilateral is the same.
 [5.3] 
Katsoulakis, M. A. & Vlachos, D. G.
From microscopic interactions to macroscopic laws of cluster evolution
postscript
abstract
We derive macroscopic governing laws of growth velocity, surface
tension, mobility, critical nucleus size, and morphological evolution
of clusters, from microscopic scale master equations for a prototype
surface reaction system with long range adsorbateadsorbate
interactions. It is shown that within the bistable regime, the
velocity of the moving cluster boundaries depends on their curvature
and is recovered from the microscopic models through derived
KuboGreen formulae.
 [5.4] 
McIntosh, I.
Harmonic tori and generalized Jacobi varieties
postscript
abstract
[From the introduction]
Over the last decade there has been considerable success in
understanding the construction of certain harmonic 2tori in symmetric
spaces (in particular, the nonsuperminimal tori in $CP^n$ and $S^n$)
using the methods of integrable systems theory. For example, if $\phi:
M\rightarrow S^2$ is a nonconformal harmonic torus one knows that
$\phi$ is determined by its (real hyperelliptic) spectral curve $X$
equipped with a degree two function $\labmda$. There are already
several ways of describing the relationship between the map \$phi$ and
its spectral data $(X,\lambda)$: see, for example, [2, 7, 12, 10]. But
we lack a direct, geometric picture of how $\phi$ arises from the
algebraic geometry of $X$. What I want to present here is such a
picture, based on a remarkable property of certain generalised Jacobi
varieties. Moreover, it is quite straightforward to extend this
picture to produce harmonic (indeed, pluriharmonic) maps into
Grassmannians and $PU_{n+1}$ (i.e. $U_{n+1}/centre) and all of these
will be maps of `finite type' in the sense of [3]. From this picture
one also sees that some of these harmonic tori are purely algebraic
and these present an interesting class for further study.
 [5.5] 
Kilian, M., McIntosh, I. & Schmitt, N.
New constant mean curvature cylinders
postscript
pdf
dvi
html
abstract
We use the DPW construction to present three new
classes of immersed CMC cylinders, each of which includes surfaces
with umbilics. The first class consists of cylinders with one end
asymptotic to a Delaunay surface. The second class presents surfaces with
a closed planar geodesic. In the third class each surface has a
closed curve of points with a common tangent plane.
 [5.6] 
Schmitt, N.
New constant mean curvature trinoids
postscript
pdf
dvi
html
abstract
I construct two families of constant mean curvature genuszero
surfaces with three ends via the DPW construction.
One of these families is known and has three unduloid ends;
the other is a new family with two unduloid and one nodoid
end.
 [5.7] 
Katsoulakis, M. A. & Kho, A.
Stochastic curvature flow: Asymptotic derivation, level set formulation and numerical experiments
postscript
abstract
We study the effects of random fluctuations included in microscopic
models for phase transitions, to macroscopic interface flows. We first
derive asymptotically a stochastic mean curvature evolution law from
the stochastic GinzburgLandau model and develop a corresponding level
set formulation. Secondly we demonstrate numerically, using stochastic
GinzburgLandau and Ising algorithms, that microscopic random
perturbations resolve geometric and numerical instabilities in the
corresponding deterministic flow.
 [5.8] 
Katsoulakis, M. A. & Vlachos, D. G.
Derivation and validation of mesoscopic theories for diffusion of interacting molecules
postscript
abstract
A mesoscopic theory for diffusion of molecules interacting with a long
range potential is derived for Arrhenius microscopic
dynamics. Gradient Monte Carlo simulations are presented on a one
dimensional lattice to assess the validity of the mesoscopic
theory. Results are compared for Metropolis and Arrhenius microscopic
dynamics. Non Fickian behavior is demonstrated and it is shown that
microscopic dynamics dictate the steadystate concentration profiles.
 [5.11] 
Horntrop, D. J., Katsoulakis, M. A. & Vlachos, D. G.
Spectral methods for mesoscopic models of pattern formation
postscript
abstract
In this paper we present spectral algorithms for the solution of
mesoscopic equations describing a broad class of pattern formation
mechanisms, focussing on a prototypical system of surface
processes. These models are in principle stochastic
integrodifferential equations and are derived directly from
microscopic lattice models, containing detailed information on
particleparticle interactions and particle dynamics. The enhanced
computational eficiency and accuracy of spectral methods versus finite
difference methods are also described.


© GANG 2001
