3,640 research outputs found
Mean Curvature Flows and Isotopy of Maps Between Spheres
Let f be a smooth map between unit spheres of possibly different dimensions.
We prove the global existence and convergence of the mean curvature flow of the
graph of f under various conditions. A corollary is that any area-decreasing
map between unit spheres (of possibly different dimensions) is homotopic to a
constant map.Comment: 21 page
Hilbert Transformation and Representation of ax+b Group
In this paper we study the Hilbert transformations over and
from the viewpoint of symmetry. For a linear operator over
commutative with the ax+b group we show that the operator is
of the form where and are the identity operator
and Hilbert transformation respectively, and are complex
numbers. In the related literature this result was proved through first
invoking the boundedness result of the operator, proved though a big machinery.
In our setting the boundedness is a consequence of the boundedness of the
Hilbert transformation. The methodology that we use is Gelfand-Naimark's
representation of the ax+b group. Furthermore we prove a similar result on the
unit circle. Although there does not exist a group like ax+b on the unit
circle, we construct a semigroup to play the same symmetry role for the Hilbert
transformations over the circle $L^2(\mathbb{T}).
Hilbert Transformation and Group
In this paper we study symmetry properties of the Hilbert transformation of
several real variables in the Clifford algebra setting. In order to describe
the symmetry properties we introduce the group which is essentially an extension of the ax+b group. The study concludes
that the Hilbert transformation has certain characteristic symmetry properties
in terms of In the present paper, for
and we obtain, explicitly, the induced spinor representations of the
group. Then we decompose the natural
representation of into the direct sum of some
two irreducible spinor representations, by which we characterize the Hilbert
transformation in and Precisely, we show that a
nontrivial skew operator is the Hilbert transformation if and only if it is
invariant under the action of the
group
On the high energy spin excitations in the high temperature superconductors: an RVB theory
The high energy spin excitation in the high T cuprates is studied in
the single mode approximation for the model. An exact form for the
mode dispersion is derived. When the Gutzwiller projected BCS state is used as
the variational ground state, a spin-wave-like dispersion of about 2.2 is
uncovered along the to line. Both the mode
energy and the integrated intensity of the spin fluctuation spectrum are found
to be almost doping independent in large doping range, which agrees very well
with the observations of recent RIXS measurements. Together with previous
studies on the quasiparticle properties of the Gutzwiiler projected BCS state,
our results indicate that such a Fermionic RVB theory can provide a consistent
description of both the itinerant and the local aspect of electronic
excitations in the high T cuprates.Comment: 7 pages, with 2.5 pages of supplementary materia
Fourier Spectrum Characterizations of Clifford Spaces on for
This article studies the Fourier spectrum characterization of functions in
the Clifford algebra-valued Hardy spaces Namely, for , Clifford algebra-valued, is
further the non-tangential boundary limit of some function in if and only if
where where the Fourier transformation and the above relation are suitably
interpreted (for some cases in the distribution sense). These results further
develop the relevant context of Alan McIntosh. As a particular case of our
results, the vector-valued Clifford Hardy space functions are identical with
the conjugate harmonic systems in the work of Stein and Weiss. The latter
proved the corresponding results in terms of the single integral form for the
cases $1\leq p<\infty.
A Bernstein type result for special Lagrangian submanifolds
Let \Sigma be a complete minimal Lagrangian submanifold of \C^n.
We identify regions in the Grassmannian of Lagrangian subspaces so that
whenever the image of the Gauss map of \Sigma lies in one of these regions,
then \Sigma is an affine space
Inverse mean curvature flows in the hyperbolic 3-space revisited
This note revisits the inverse mean curvature flow in the 3-dimensional
hyperbolic space. In particular, we show that the limiting shape is not
necessarily round after scaling, thus resolving an inconsistency in the
literature.Comment: The higher dimensional case is added. To appear in Calculus of
Variations and PDE'
Curvature Decay Estimates of Graphical Mean Curvature Flow in Higher Codimensions
We derive pointwise curvature estimates for graphical mean curvature flows in
higher codimensions. To the best of our knowledge, this is the first such
estimates without assuming smallness of first derivatives of the defining map.
An immediate application is a convergence theorem of the mean curvature flow of
the graph of an area decreasing map between flat Riemann surfaces.Comment: arXiv admin note: text overlap with arXiv:math/030224
An efficient Monte Carlo algorithm for the evaluation of Renyi entanglement entropy of a general quantum dimer model at the R-K point
A highly efficient and simple to implement Monte Carlo algorithm is proposed
for the evaluation of the Renyi entanglement entropy(REE) of quantum dimer
model(QDM) at the Rokhsar-Kivelson(R-K) point. It makes possible the evaluation
of REE at the R-K point to the thermodynamic limit for a general QDM. We apply
the algorithm to QDM on both triangular and square lattice as demonstrations
and find the REE on both lattices follow perfect linear scaling in the
thermodynamic limit, apart from an even-odd oscillation in the latter case. We
also evaluate the topological entanglement entropy(TEE) on both lattices with a
subtraction procedure. While the expected TEE of is clearly demonstrated
for QDM on triangular lattice, a strong oscillation of the result is found for
QDM on square lattice, which implies the relevance of boundary perturbation in
such a critical system.Comment: 6 page
Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities
In this paper, we study solutions to the linearized vacuum Einstein equations
centered at higher-dimensional Schwarzschild met- rics. We employ Hodge
decomposition to split solutions into scalar, co-vector, and two-tensor pieces;
the first two portions respectively cor- respond to the closed and co-closed,
or polar and axial, solutions in the case of four spacetime dimensions, while
the two-tensor portion is a new feature in the higher-dimensional setting.
Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge
decomposition, we produce decoupled gauge-invariant master quantities
satisfying Regge- Wheeler type wave equations in each of the three portions.
The scalar and co-vector quantities respectively generalize the
Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four
spacetime dimen- sions; beyond these quantities, we further discover a
higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the
co-vector por- tion. In the analysis of the master quantities, we strengthen
the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate
in all dimensions; further, we prove decay estimates in the case of six or
fewer spacetime dimensions. Finally, we provide a rigorous argument that
linearized solutions of low angular frequency are decomposable as a sum of pure
gauge solution and linearized Myers-Perry solution, the lat- ter solutions
generalizing the linearized Kerr solutions in four spacetime dimensions.Comment: 71 page
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