Remarks on the extension of the Ricci flow

We present two new conditions to extend the Ricci flow on a compact manifold over a finite time, which are improvements of some known extension theorems.Comment: 9 pages, to appear in Journal of Geometric Analysi

Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary

Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with smooth n-1 dimensional boundary. We search the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary conditions or for the singularly perturbed Klein Gordon Maxwell system with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that stable critical points of the mean curvature of the boundary generates solutions when the perturbation parameter is sufficiently small.Comment: arXiv admin note: text overlap with arXiv:1410.884

Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrodinger system

In this paper, we establish two new types of invariant sets for the coupled nonlinear Schrodinger system on $\mathbb{R}^n$, and derive two sharp thresholds of blow-up and global existence for its solution. Some analogous results for the nonlinear Schrodinger system posed on the hyperbolic space $\mathbb{H}^n$ and on the standard 2-sphere $\mathbb{S}^2$ are also presented. Our arguments and constructions are improvements of some previous works on this direction. At the end, we give some heuristic analysis about the strong instability of the solitary waves.Comment: 21 page

Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics

In this article we develop some new existence results for the Einstein constraint equations using the Lichnerowicz-York conformal rescaling method. The mean extrinsic curvature is taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant. The rescaled background metric belongs to the positive Yamabe class, and the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are taken to be sufficiently small, with the matter energy density not identically zero. Using topological fixed-point arguments and global barrier constructions, we then establish existence of solutions to the constraints. Two recent advances in the analysis of the Einstein constraint equations make this result possible: A new type of topological fixed-point argument without smallness conditions on spatial derivatives of the mean extrinsic curvature, and a new construction of global super-solutions for the Hamiltonian constraint that is similarly free of such conditions on the mean extrinsic curvature. For clarity, we present our results only for strong solutions on closed manifolds. However, our results also hold for weak solutions and for other cases such as compact manifolds with boundary; these generalizations will appear elsewhere. The existence results presented here for the Einstein constraints are apparently the first such results that do not require smallness conditions on spatial derivatives of the mean extrinsic curvature.Comment: 4 pages, no figures, accepted for publication in Physical Review Letters. (Abstract shortenned and other minor changes reflecting v4 version of arXiv:0712.0798

Positive mass theorem for the Paneitz-Branson operator

We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert.Comment: 7 page

Heat flow method to Lichnerowicz type equation on closed manifolds

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solutions to the following heat equation {ll} u_t-\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad in\quad M with the positive smooth initial data $u_0$.Comment: 10 page

On the master equation approach to kinetic theory: linear and nonlinear Fokker--Planck equations

We discuss the relationship between kinetic equations of the Fokker-Planck type (two linear and one non-linear) and the Kolmogorov (a.k.a. master) equations of certain N-body diffusion processes, in the context of Kac's "propagation of chaos" limit. The linear Fokker-Planck equations are well-known, but here they are derived as a limit N->infty of a simple linear diffusion equation on (3N-C)-dimensional N-velocity spheres of radius sqrt(N) (with C=1 or 4 depending on whether the system conserves energy only or energy and momentum). In this case, a spectral gap separating the zero eigenvalue from the positive spectrum of the Laplacian remains as N->infty,so that the exponential approach to equilibrium of the master evolution is passed on to the limiting Fokker-Planck evolution in R^3. The non-linear Fokker-Planck equation is known as Landau's equation in the plasma physics literature. Its N-particle master equation, originally introduced (in the 1950s) by Balescu and Prigogine (BP), is studied here on the (3N-4)-dimensional N-velocity sphere. It is shown that the BP master equation represents a superposition of diffusion processes on certain two-dimensional sub-manifolds of R^{3N} determined by the conservation laws for two-particle collisions. The initial value problem for the BP master equation is proved to be well-posed and its solutions are shown to decay exponentially fast to equilibrium. However, the first non-zero eigenvalue of the BP operator is shown to vanish in the limit N->infty. This indicates that the exponentially fast approach to equilibrium may not be passed from the finite-N master equation on to Landau's nonlinear kinetic equation.Comment: 20 pages; based on talk at the 18th ICTT Conference. Some typos and a few minor technical fixes. Modified title slightl

Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension N \geg 3, we are interested in the critical equation $$\Delta_g u+(N-2/4(N-1) S_g+\epsilon h)u=u^{N+2/N-2} in M, u>0 in M,$$ where \Delta_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), $h\in C^{0,\alpha}(M)$, and $\epsilon$ is a small parameter

Conformal scattering for a nonlinear wave equation on a curved background

The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple spacetime. The scattering operator is obtained via trace operators at null infinities. The proof is achieved in three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field in the Schwarzschild spacetime and a method used by H\"ormander for the Goursat problem. A well-posedness result for the characteristic Cauchy problem on a light cone at infinity is then obtained. This requires a control of the nonlinearity uniform in time which comes from an estimates of the Sobolev constant and a decay assumption on the nonlinearity of the equation. Finally, the trace operators on conformal infinities are built and used to define the conformal scattering operator

A model problem for conformal parameterizations of the Einstein constraint equations

We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by taking the quotient of certain symmetric data on conformally flat tori. Specializing the model problem to a three-parameter family of conformal data we observe a number of new phenomena for the conformal and CTS methods. Within this family, we obtain a general existence theorem so long as the mean curvature does not change sign. When the mean curvature changes sign, we find that for certain data solutions exist if and only if the transverse-traceless tensor is sufficiently small. When such solutions exist, there are generically more than one. Moreover, the theory for mean curvatures changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.Comment: 40 pages, 4 figure