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 $L^2(\mathbb{R})$ and $L^2(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$ commutative with the ax+b group we show that the operator is of the form $\lambda I+\eta H,$ where $I$ and $H$ are the identity operator and Hilbert transformation respectively, and $\lambda,\eta$ 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}).

On the high energy spin excitations in the high temperature superconductors: an RVB theory

The high energy spin excitation in the high T$_{c}$ cuprates is studied in the single mode approximation for the $t-t'-J$ 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$J$ is uncovered along the $\Gamma=(0,0)$ to $\mathrm{M}=(\pi,0)$ 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$_{c}$ cuprates.Comment: 7 pages, with 2.5 pages of supplementary materia

Fourier Spectrum Characterizations of Clifford HpH^{p} Spaces on R+n+1\mathbf{R}^{n+1}_+ for 1≤p≤∞1\leq p \leq \infty

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces $H^p(\mathbf R^{n+1}_+), 1\leq p\leq \infty.$ Namely, for $f\in L^p(\mathbf R^n)$, Clifford algebra-valued, $f$ is further the non-tangential boundary limit of some function in $H^p(\mathbf R^{n+1}_+),$ $1\leq p\leq \infty,$ if and only if $\hat{f}=\chi_+\hat{f},$ where $\chi_+(\underline{\xi})=\frac{1}{2}(1+i\frac{\underline \xi}{|\underline \xi|}),$ 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.

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 $\ln2$ 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

Generalized Lagrangian mean curvature flows: the cotangent bundle case

In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost K\"ahler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n, 0) form, just like the Calabi-Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.Comment: 31 page

Topological entanglement entropy in Gutzwiller projected spin liquids

The topological entanglement entropy(TEE) of Gutzwiller projected RVB state is studied with Monte Carlo simulation. New tricks are proposed to improve the convergence of TEE, which enable us to show that the spin liquid state studied in Ref.\cite{Vishwanath} actually does not support $Z_{2}$ topological order, a conclusion that is consistent with the information drawn from the inspection of the topological degeneracy on the same state. We find both a long ranged RVB amplitude and an approximate Marshall sign structure are at the origin of the suppression of vison gap in this spin liquid state. On the other hand, robust signature of $Z_{2}$ topological order, i.e., a TEE of $\ln2$, is clearly demonstrated for a Gutzwiiler projected RVB state on triangular lattice which is evolved from the RVB state proposed originally by Anderson\cite{Sorella}. We also find that it is the sign, rather than the amplitude of the RVB wave function, that dominates the TEE and is responsible for a positive value of TEE, which implies that the nonlocal entanglement in the RVB state is mainly encoded in the sign of the RVB wave function. Our results indicate that some information that is important for the topological property of a RVB state is missed in the effective field theory description and that a $Z_{2}$ gauge structure in the saddle point action is not enough for the RVB state to exhibit topological order, even if its spin correlation is extremely short ranged.Comment: 9 page

High order maximum principle preserving finite volume method for convection dominated problems

In this paper, we investigate the application of the maximum principle preserving (MPP) parametrized flux limiters to the high order finite volume scheme with Runge-Kutta time discretization for solving convection dominated problems. Such flux limiter was originally proposed in [Xu, Math. Comp., 2013] and further developed in [Xiong et. al., J. Comp. Phys., 2013] for finite difference WENO schemes with Runge-Kutta time discretization for convection equations. The main idea is to limit the temporal integrated high order numerical flux toward a first order MPP monotone flux. In this paper, we generalize such flux limiter to high order finite volume methods solving convection-dominated problems, which is easy to implement and introduces little computational overhead. More importantly, for the first time in the finite volume setting, we provide a general proof that the proposed flux limiter maintains high order accuracy of the original WENO scheme for linear advection problems without any additional time step restriction. For general nonlinear convection-dominated problems, we prove that the proposed flux limiter introduces up to $O(\Delta x^3+\Delta t^3)$ modification to the high order temporal integrated flux in the original WENO scheme without extra time step constraint. We also numerically investigate the preservation of up to ninth order accuracy of the proposed flux limiter in a general setting. The advantage of the proposed method is demonstrated through various numerical experiments