176 research outputs found
Nematic Films and Radially Anisotropic Delaunay Surfaces
We develop a theory of axisymmetric surfaces minimizing a combination of
surface tension and nematic elastic energies which may be suitable for
describing simple film and bubble shapes. As a function of the elastic constant
and the applied tension on the bubbles, we find the analogues of the unduloid,
sphere, and nodoid in addition to other new surfaces.Comment: 15 pages, 18 figure
Coherent Beam-Beam Tune Shift of Unsymmetrical Beam-Beam Interactions with Large Beam-Beam Parameter
Coherent beam-beam tune shift of unsymmetrical beam-beam interactions was
studied experimentally and numerically in HERA where the lepton beam has a very
large beam-beam parameter (up to ). Unlike the symmetrical case of
beam-beam interactions, the ratio of the coherent and incoherent beam-beam tune
shift in this unsymmetrical case of beam-beam interactions was found to
decrease monotonically with increase of the beam-beam parameter. The results of
self-consistent beam-beam simulation, the linearized Vlasov equation, and the
rigid-beam model were compared with the experimental measurement. It was found
that the coherent beam-beam tune shifts measured in the experiment and
calculated in the simulation agree remarkably well but they are much smaller
than those calculated by the linearized Vlasov equation with the single-mode
approximation or the rigid-beam model. The study indicated that the single-mode
approximation in the linearization of Vlasov equation is not valid in the case
of unsymmetrical beam-beam interactions. The rigid-beam model is valid only
with a small beam-beam parameter in the case of unsymmetrical beam-beam
interactions.Comment: 32 pages, 13 figure
Special biconformal changes of K\"ahler surface metrics
The term "special biconformal change" refers, basically, to the situation
where a given nontrivial real-holomorphic vector field on a complex manifold is
a gradient relative to two K\"ahler metrics, and, simultaneously, an
eigenvector of one of the metrics treated, with the aid of the other, as an
endomorphism of the tangent bundle. A special biconformal change is called
nontrivial if the two metrics are not each other's constant multiples. For
instance, according to a 1995 result of LeBrun, a nontrivial special
biconformal change exists for the conformally-Einstein K\"ahler metric on the
two-point blow-up of the complex projective plane, recently discovered by Chen,
LeBrun and Weber; the real-holomorphic vector field involved is the gradient of
its scalar curvature. The present paper establishes the existence of nontrivial
special biconformal changes for some canonical metrics on Del Pezzo surfaces,
viz. K\"ahler-Einstein metrics (when a nontrivial holomorphic vector field
exists), non-Einstein K\"ahler-Ricci solitons, and K\"ahler metrics admitting
nonconstant Killing potentials with geodesic gradients.Comment: 16 page
Bounding λ2 for KaÌhlerâEinstein metrics with large symmetry groups
We calculate an upper bound for the second non-zero eigenvalue of the scalar Laplacian, λ2, for toric-KaÌhlerâEinstein metrics in terms of the polytope data. We also give a similar upper bound for KoisoâSakane type KaÌhlerâEinstein metrics. We provide some detailed examples in complex dimensions 1, 2 and 3
Very Singular Diffusion Equations-Second and Fourth Order Problems
This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an Hâ1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example
Stability of the selfsimilar dynamics of a vortex filament
In this paper we continue our investigation about selfsimilar solutions of
the vortex filament equation, also known as the binormal flow (BF) or the
localized induction equation (LIE). Our main result is the stability of the
selfsimilar dynamics of small pertubations of a given selfsimilar solution. The
proof relies on finding precise asymptotics in space and time for the tangent
and the normal vectors of the perturbations. A main ingredient in the proof is
the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger
equation, connected to the binormal flow by Hasimoto's transform.Comment: revised version, 36 page
Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability
This paper concerns the explicit construction of extremal Kaehler metrics on
total spaces of projective bundles, which have been studied in many places. We
present a unified approach, motivated by the theory of hamiltonian 2-forms (as
introduced and studied in previous papers in the series) but this paper is
largely independent of that theory.
We obtain a characterization, on a large family of projective bundles, of
those `admissible' Kaehler classes (i.e., the ones compatible with the bundle
structure in a way we make precise) which contain an extremal Kaehler metric.
In many cases, such as on geometrically ruled surfaces, every Kaehler class is
admissible. In particular, our results complete the classification of extremal
Kaehler metrics on geometrically ruled surfaces, answering several
long-standing questions.
We also find that our characterization agrees with a notion of K-stability
for admissible Kaehler classes. Our examples and nonexistence results therefore
provide a fertile testing ground for the rapidly developing theory of stability
for projective varieties, and we discuss some of the ramifications. In
particular we obtain examples of projective varieties which are destabilized by
a non-algebraic degeneration.Comment: 40 pages, sequel to math.DG/0401320 and math.DG/0202280, but largely
self-contained; partially replaces and extends math.DG/050151
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
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