


Document Library: Series 4 Abstracts

[4.1] 
Kusner, R., Mazzeo, R. & Pollack, D.
The moduli space of complete embedded constant mean curvature surfaces
postscript
abstract
We examine the space of surfaces in R^{3}
which are complete, properly
embedded and have nonzero constant mean curvature. These surfaces are
noncompact provided we exclude the case of the round sphere. We prove that
the space $M_{k}$ of all such surfaces with
$k$ ends (where surfaces are identified if they differ by an isometry of
R^{3})is locally a finite dimensional real
analytic variety. When the linearization of the quasilinear elliptic
equation specifying mean curvature equal to one has no
L^{2}nullspace we prove that
$M_{k}$ is locally the quotient of a
real analytic manifold of dimension 3$k$6 by a finite group (i.e.,
a real analytic orbifold), for $k>2$. This finite group is the isotropy
subgroup of the surface in the group of Euclidean motions. It is of interest
to note that the dimension of $M_{k}$ is
independent of the topology of the underlying punctured Riemann surface to
which SIGMA is conformally equivalent. These results also apply to
hypersurfaces of H^{n+1} with nonzero
constant mean curvature greater than that of a horosphere and whose ends are
cylindrically bounded.
 [4.2] 
Kusner, R., Lahti, P. & Lillya, C. P.
New surface allotropes of carbon
postscript
pdf
dvi
abstract
The simplest negatively curved (schwartzene) surface allotrope of carbon,
$C_{32}$trousers is a stable unit and also
demonstrate the viability of several related schwartzenes. In contrast to
many schwartzene structures proposed earlier, $C_{32}$ is a reasonable
target for organic synthesis.
 [4.4] 
Burstall, F. E. & Pedit, F.
Dressing orbits of harmonic maps
postscript
pdf
dvi
abstract
[From the introduction]
At the heart of the modern theory of harmonic maps from a Riemann
surface to a Riemannian symmetric space is the observation that, in
this setting, the harmonic map equations have a zero curvature
representation [19, 24, 28] and so correspond to loops of flat
connections. This fact was first exploited in the mathematical
literature by Uhlenbeck in her study [24] of harmonic maps $R\leftarrow G$
into a compact Lie group $G$. Uhlenbeck discovered that harmonic maps
correspond to certain holomorphic maps, the extended solutions, into
the based loop group $\Omega G$ and used this to define an action of a
certain loop group on the space of harmonic maps. However, the main
focus of [24] was on harmonic maps of a 2sphere and for these maps
the action reduces to an action of a finitedimensional quotient group
(see also [1, 9]).
 [4.16] 
HertrichJeromin, U. & Pedit, F.
Remarks on the darboux transform of isothermic surfaces
postscript
pdf
dvi
abstract
We study Darboux and Christoffel transforms of isothermic surfaces in Euclidean
space. The transformations play a significant role in relation to integrable
system theory. Surfaces of constant mean curvature turn out to be special
among all isothermic surfaces: Their parallel constant mean curvature surfaces
are Christoffel and Darboux transforms at the same time. We prove  as a
generalization of Bianchi's theorem on minimal Darboux transforms of minimal
surfaces  that constant mean curvature surfaces in Euclidean space allow
infinity ^3 Darboux transforms into surfaces of constant mean
curvature. We indicate the connection of these Darboux transforms to
Bäcklund transforms of spherical surfaces.
 [4.18] 
Kusner, R. & Schmitt, N.
The spinor representation of surfaces in space
postscript
abstract
[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.
 [4.19] 
Katsoulakis, M. A. & Souganidis, P. E.
Stochastic Ising models and anisotropic front propagation
postscript
pdf
dvi
abstract
We study Ising models with general spin flip dynamics obeying the detailed
balance law. After passing to suitable macroscopic limits, we obtain interfaces
moving with normal velocity depending anisotropically on their principal
curvatures and direction. In addition we deduce (directiondependent) Kubo
Greentype formulae for the mobility and the hessian of the surface tension,
thus obtaining an explicit description of anisotropy in terms of microscopic
quantities. The choid of dynamics affects only the mobility, a scalar function
of the direction.
 [4.20] 
GrosseBrauckmann, K. & Kusner, R.
Moduli spaces of embedded constant mean curvature surfaces with few ends and special symmetry
postscript
abstract
We give sharp, necessary conditions on complete embedded CMC surfaces with
three of four ends subject to reflection symmetries. The respective submoduli
spaces are twodimensional varieties in the moduli spaces of general CMC
surfaces. Fundamental domains of our CMC surfaces are characterized by
associated great circle polygons in the threesphere.
 [4.21] 
Kamberov, G., Pedit, F. & Pinkall, U.
Bonnet pairs and isothermic surfaces
postscript
pdf
dvi
abstract
[From the introduction]
A classical question in surface theory is which data are sufficient to
describe a surface in space up to rigid motions. Bonnet suggested that
mean curvature and metric should suffice to determine the surface
generically. The local theory (without umbilic points) was developed
by Bonnet [6], Cartan [4] and Chern [5] who showed the existence of
various 1parameter families of Bonnet surfaces, i.e., surfaces with
the same induced metric and mean curvature. A comprehensive study of
this problem and its relationship to the Painleve equations has
recently been completed by Bobenko and Eitner [2]. On the other hand,
Lawson and Tribuzy [9] have shown that for embedded compact surfaces
there are at most two surfaces to a given metric and mean
curvature. Moreover, uniqueness can be established under various
global assumptions [8]. Up to date it is unknown whether such compact
Bonnet pairs exist and if, how to construct them.
 [4.24] 
Kusner, R. & Sullivan, J.
On distortion and thickness of knots
postscript
abstract
[From the introduction]
What length of rope (of given diameter) is required to tie a
particular knot? Or, to turn the problem around, given an embedded
curve, how thick a regular neighborhood of the curve also is embedded?
Intuitively, the diameter of the possible rope is bounded by the
distance between strands at the closest crossing in the knot. But of
course the distance between two points along a curve goes to zero as
the points approach each other, so to make the notion precise, we need
to exclude some neighborhood of the diagonal.
 [4.26] 
Jin, S. & Katsoulakis, M. A.
Relaxation approximations to front propagation
postscript
pdf
dvi
abstract
We introduce a relaxation model for front propagation problems. Our proposed
relaxation approximation is a semilinear hyperbolic system without
singularities. It yields a directiondependent normal velocity at the leading
term and captures, in the ChapmanEnskog expansion, the higher order curvature
dependent corrections, including possible anisotropies.
 [4.29] 
GrosseBrauckmann, K., Kusner, R. & Sullivan, J.
Classification of embedded constant mean curvature surface with genus zero and three ends
postscript
abstract
For each embedded constant mean curvature surface in
R^{3} with
three ends and genus zero, we construct a conjugate cousin boundary
contour in S^{3}. The moduli space
of such contours is parametrized by
the space of triangles on S^{2}. This
imposes necessary conditions on these CMC surfaces, and we expect this
space of triangles (up to rotation of S^{2})
exactly parametrizes the moduli space of these
surfaces (up to rigid motion of R^{3}).
Remarks on the implications for CMC surfaces with more ends and higher genus
are also included.
 [4.30] 
Katsoulakis, M. A., Kossioris, G. & Makridakis, C.
Convergence and error estimates of relaxation schemes for multidimensional conservation laws
postscript
pdf
dvi
abstract
We study discrete and semidiscrete relaxation schemes for multidimensional
scalar conservation laws. We show convergence of the relaxation schemes to
the entropy solution of the conservation law and derive error estimates
that exhibit the precise interaction between the relaxation time and the
space/time discretization parameters of these schemes.
 [4.31] 
GrosseBrauckmann, K., Kusner, R. & Sullivan, J.
Constant mean curvature surfaces with cylindrical ends
postscript
pdf
dvi
abstract
R. Schoen has asked whether the sphere and the cylinder are the only complete
(almost) embedded constant mean curvature surfaces with finite absolute
total curvature. We propose an infinite family of such surfaces. The
existence of examples of this kind is supported by results of computer
experiments carried out using an algorithm developed by Oberknapp and Polthier.
 [4.32] 
HertrichJeromin, U.
The surfaces capable of division into infinitesimal squares by their curves of curvature
abstract
Classically, isothermic surfaces are characterized as those surfaces which
"are divisible into infinitesimal squares by their curvature lines". This
characterization is the direct analogue to the definition of discrete
isothermic nets. In order to understand the relations between the discrete and
smooth theory better, it is described how to give the classical
characterizations a rigorous meaning in the sense of modern differential
geometry.
 [4.33] 
HertrichJeromin, U.
Supplement on curved flats in the space of point pairs and isothermic surfaces: a quaternionic calculus
abstract
A quaternionic calculus for surface pairs in the conformal 4sphere is
elaborated. This calculus is then used to discuss the relation between
curved flats in the symmetric space of pair points and Darboux and
Christoffel pairs of isothermic surfaces/ A new viewpoint on relations
between surfaces of constant mean curvature in certain space forms is
presented  in particular, a new form of Bryant's Weierstrasstype
representation for surfaces of constant mean curvature 1 in hyperbolic
3space is given.


© GANG 2001
