2,020 research outputs found
Caccioppoli's inequalities on constant mean curvature hypersurfaces in Riemannian manifolds
This is a revised version (minor changes and a deeper insight in the positive
curvature case).
We prove some Caccioppoli's inequalities for the traceless part of the second
fundamental form of a complete, noncompact, finite index, constant mean
curvature hypersurface of a Riemannian manifold, satisfying some curvature
conditions. This allows us to unify and clarify many results scattered in the
literature and to obtain some new results. For example, we prove that there is
no stable, complete, noncompact hypersurface in
with constant mean curvature provided that, for suitable the
-norm of the traceless part of second fundamental form satisfies some
growth condition.Comment: 31 page
Some properties of simple minimal knots
A minimal knot is the intersection of a topologically embedded branched
minimal disk in with a small sphere centered at
the branch point. When the lowest order terms in each coordinate component of
the embedding of the disk in are enough to determine the knot
type, we talk of a simple minimal knot. Such a knot is given by three integers
; denoted by , it can be parametrized in the cylinder as
. From this
expression stems a natural representation of as an -braid. In
this paper, we give a formula for its writhe number, i.e. the signed number of
crossing points of this braid and derive topological consequences. We also show
that if and are not mutually prime, is periodic. Simple
minimal knots are a generalization of torus knots
Stochastic acceleration in a random time-dependent potential
We study the long time behaviour of the speed of a particle moving in
under the influence of a random time-dependent potential
representing the particle's environment. The particle undergoes successive
scattering events that we model with a Markov chain for which each step
represents a collision. Assuming the initial velocity is large enough, we show
that, with high probability, the particle's kinetic energy grows as
when
Some existence results to the Dirichlet problem for the minimal hypersurface equation on non mean convex domains of a Riemannian manifold
We prove the existence of minimal hypersurfaces for the Dirichlet that
extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian
ambient manifold
Examples of scalar-flat hypersurfaces in
Given a hypersurface of null scalar curvature in the unit sphere
, , such that its second fundamental form has rank
greater than 2, we construct a singular scalar-flat hypersurface in \Rr^{n+1}
as a normal graph over a truncated cone generated by . Furthermore, this
graph is 1-stable if the cone is strictly 1-stable.Comment: Paper accepted to publication in Manuscripta Mathematic
Chaotic dynamics of superconductor vortices in the plastic phase
We present numerical simulation results of driven vortex lattices in presence
of random disorder at zero temperature. We show that the plastic dynamics is
readily understood in the framework of chaos theory. Intermittency "routes to
chaos" have been clearly identified, and positive Lyapunov exponents and
broad-band noise, both characteristic of chaos, are found to coincide with the
differential resistance peak. Furthermore, the fractal dimension of the strange
attractor reveals that the chaotic dynamics of vortices is low-dimensional.Comment: 5 pages, 3 figures Accepted for publication in Physical Review
Letter
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