5,587 research outputs found
Spherical Surfaces
We study surfaces of constant positive Gauss curvature in Euclidean 3-space
via the harmonicity of the Gauss map. Using the loop group representation, we
solve the regular and the singular geometric Cauchy problems for these
surfaces, and use these solutions to compute several new examples. We give the
criteria on the geometric Cauchy data for the generic singularities, as well as
for the cuspidal beaks and cuspidal butterfly singularities. We consider the
bifurcations of generic one parameter families of spherical fronts and provide
evidence that suggests that these are the cuspidal beaks, cuspidal butterfly
and one other singularity. We also give the loop group potentials for spherical
surfaces with finite order rotational symmetries and for surfaces with embedded
isolated singularities.Comment: 23 pages, 18 figures. Version 3: Typos correcte
Local Isometric immersions of pseudo-spherical surfaces and evolution equations
The class of differential equations describing pseudo-spherical surfaces,
first introduced by Chern and Tenenblat [3], is characterized by the property
that to each solution of a differential equation, within the class, there
corresponds a 2-dimensional Riemannian metric of curvature equal to . The
class of differential equations describing pseudo-spherical surfaces carries
close ties to the property of complete integrability, as manifested by the
existence of infinite hierarchies of conservation laws and associated linear
problems. As such, it contains many important known examples of integrable
equations, like the sine-Gordon, Liouville and KdV equations. It also gives
rise to many new families of integrable equations. The question we address in
this paper concerns the local isometric immersion of pseudo-spherical surfaces
in from the perspective of the differential equations that give
rise to the metrics. Indeed, a classical theorem in the differential geometry
of surfaces states that any pseudo-spherical surface can be locally
isometrically immersed in . In the case of the sine-Gordon
equation, one can derive an expression for the second fundamental form of the
immersion that depends only on a jet of finite order of the solution of the
pde. A natural question is to know if this remarkable property extends to
equations other than the sine-Gordon equation within the class of differential
equations describing pseudo-spherical surfaces. In an earlier paper [11], we
have shown that this property fails to hold for all other second order
equations, except for those belonging to a very special class of evolution
equations. In the present paper, we consider a class of evolution equations for
of order describing pseudo-spherical surfaces. We show that
whenever an isometric immersion in exists, depending on a jet of
finite order of , then the coefficients of the second fundamental forms are
functions of the independent variables and only.Comment: Fields Institute Communications, 2015, Hamiltonian PDEs and
Applications, pp.N
Families of spherical surfaces and harmonic maps
We study singularities of constant positive Gaussian curvature surfaces and
determine the way they bifurcate in generic 1-parameter families of such
surfaces. We construct the bifurcations explicitly using loop group methods.
Constant Gaussian curvature surfaces correspond to harmonic maps, and we
examine the relationship between the two types of maps and their singularities.
Finally, we determine which finitely A-determined map-germs from the plane to
the plane can be represented by harmonic maps.Comment: 30 pages, 7 figures. Version 2: substantial revision compared with
version 1. The results are essentially the same, but some of the arguments
are improved or correcte
Green's functions of potential problems in lens shaped geometries
The Kelvin inversion method will be outlined to determine the potential distribution due to a point charge (or the Green's function) in geometries bounded by flat and spherical surfaces
Fresnel cup reflector directs maximum energy from light source
To minimize shielding and overheating, a composite Fresnel cup reflector design directs the maximum energy from a light source. It consists of a uniformly ellipsoidal end surface and an extension comprising a series of confocal ellipsoidal and concentric spherical surfaces
Phase transition of triangulated spherical surfaces with elastic skeletons
A first-order transition is numerically found in a spherical surface model
with skeletons, which are linked to each other at junctions. The shape of the
triangulated surfaces is maintained by skeletons, which have a one-dimensional
bending elasticity characterized by the bending rigidity , and the surfaces
have no two-dimensional bending elasticity except at the junctions. The
surfaces swell and become spherical at large and collapse and crumple at
small . These two phases are separated from each other by the first-order
transition. Although both of the surfaces and the skeleton are allowed to
self-intersect and, hence, phantom, our results indicate a possible phase
transition in biological or artificial membranes whose shape is maintained by
cytoskeletons.Comment: 15 pages with 10 figure
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