We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $ x ^{- ρ}$ with $ρ>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $ρ> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $ρ>0$ for $d= 3$.
We prove a local support theorem for the radiation fields on asymptotically Euclidean manifolds that partly generalizes the local support theorem for the Radon transform.
In this paper we verify the Strauss conjecture for semilinear wave equations on asymptotically Euclidean manifolds when n=3,4, we also give an almost sharp life span for the subcritical case $2\le p<p_c$ when n=3. The main ingredients include a KSS type estimate with $0<μ<1/2$ and weighted Strichartz estimates of order two.
In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with time dependent damping and potential, and mixed nonlinearities $c_1 |u_t|^p+c_2 |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and the Glassey conjecture.
For Laplace-Beltrami operators associated to metrics which are long range perturbations of the flat one, we prove estimates for powers of the resolvent as the spectral parameter goes to zero. We also discuss applications to the local energy decay for the Schroedinger, Wave and Klein Gordon equations.
We develop the notion of Lagrangian distribution on scattering manifolds, meaning on the compactified cotangent bundle, which is a manifold with corners equipped with a scattering symplectic structure. In particular, we study the notion of principal symbol of the arising class of distributions.
The purpose of this article is twofold. First we give a very robust method for proving sharp time decay estimates for the most classical three models of dispersive Partial Differential Equations, the wave, Klein-Gordon and Schr{ö}dinger equations, on curved geometries, showing under very general assumptions the exact same decay as for the Euclidean case. Then we also extend these decay properties to the case of boundary value problems.
We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on $(R^3, g)$, where the metric $g$ is a small perturbation of the flat metric and approaches the Euclidean metric like $(1+|x|)^{-a}$ with $a>1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in the appendix.
In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and Glassey conjecture. In some cases, we obtain existence results, where the lower bound of the lifespan agrees with the upper bound in order. In addition, our results apply for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign co…
We present a method in nonlinear elliptic systems to study curvature decays on asymptotically locally Euclidean (ALE) manifolds. In particular, we show that scalar flat Kahler and harmonic ALE metrics of real dimension n are of order n-2.
In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<p\le p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<p\le 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties is that we can deal with the problem with flat metric…
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also s…
In this note we prove two existence theorems for the Einstein constraint equations on asymptotically Euclidean manifolds. The first is for arbitrary mean curvature functions with restrictions on the size of the transverse-traceless data and the non-gravitational field data, while the second assumes a near-CMC condition, with no other restrictions.
We show improved local energy decay for the wave equation on asymptotically Euclidean manifolds in odd dimensions in the short range case. The precise decay rate depends on the decay of the metric towards the Euclidean metric. We also give estimates of powers of the resolvent of the wave propagator between weighted spaces.
We show that any asymptotically locally Euclidean (ALE) metric which is obstruction-flat or extended obstruction-flat must be ALE of a certain optimal order. Moreover, our proof applies to very general elliptic systems and in any dimension $n \geq 3$. The proof is based on the technique of Cheeger-Tian for Ricci-flat metrics. We also apply this method to obtain a singularity removal theorem for (extended) obstruction-flat metrics with isolated $C^0$-orbifold singular points.
In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean metric. One of the main ingredients in our proof is the construction of (unbounded) positive entire solutions for $Δ_{g}ϕ_λ=λ^{2}ϕ_λ$, with certain estimates which are uniform for small parameter $λ\in (0,λ_0)$. In addition, our argument works equally well for semilinear damped w…
In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely $e^{-itΛ_g^σ}$ with $σ\in (0,\infty)\backslash \{1\}$ and $Λ_g:=\sqrt{-Δ_g}$ where $Δ_g$ is the Laplace-Beltrami operator on asymptotically Euclidean manifolds $(\mathbb{R}^d,g)$. Let $f_0\in C^\infty_0(\mathbb{R})$ be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part $(1-f_0)(P)e^{-itΛ_g^σ}$ satisfies global in time Strichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 2$ inside a compact set under non-trapping condition. On the other hand, under t…
In this note, we shall prove geodesic convexity of the space of Kähler potentials on an ALE Kähler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness from below of the Mabuchi energy, and give an alternative proof for the uniqueness of scalar-flat metrics in the category of ALE Kähler manifold.
We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant $Y\leq 0$. Previous work by the second and third named authors \cite{ChenWang} showed that while the Yamabe flow always converges in a global weighted sense when $Y>0$, the flow must diverge when $Y\leq 0$. We show here in the $Y\leq 0$ case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.
We verify the three-dimensional Glassey conjecture for exterior domain (M, g), where the metric g is asymptotically Euclidean, provided that certain local energy assumption is satisfied. The radial Glassey conjecture exterior to a ball is also verified for dimension three or higher. The local energy assumption is satisfied for many important cases, including exterior domain with nontrapping obstacles and flat metric, exterior domain with star-shaped obstacles and small asymptotically Euclidean metric, as well as the nontrapping asymptotically Euclidean manifolds (R^n, g).
This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data s…
We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order $(1,1)$, on an asymptotically Euclidean manifold. We first prove a two term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the co…
Let P be a long range metric perturbation of the Euclidean Laplacian on R^d, d>1. We prove local energy decay for the solutions of the wave, Klein-Gordon and Schroedinger equations associated to P. The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group exp(itf(P)) where f has a suitable development at zero (resp. infinity).
In this article we further develop the solution theory for the Einstein constraint equations on an n-dimensional, asymptotically Euclidean manifold M with interior boundary S. Building on recent results for both the asymptotically Euclidean and compact with boundary settings, we show existence of far-from-CMC and near-CMC solutions to the conformal formulation of the Einstein constraints when nonlinear Robin boundary conditions are imposed on S, similar to those analyzed previously by Dain (2004), by Maxwell (2004, 2005), and by Holst and Tsogtgerel (2013) as a model of black holes in various …
We consider the resolvent on asymptotically Euclidean warped product manifolds in an appropriate 0-Gevrey class, with trapped sets consisting of only finitely many components. We prove that the high-frequency resolvent is either bounded by $C_ε|λ|^ε$ for any $ε>0$, or blows up faster than any polynomial (at least along a subsequence). A stronger result holds if the manifold is analytic. The method of proof is to exploit the warped product structure to separate variables, obtaining a one-dimensional semiclassical Schrödinger operator. We then classify the microlocal resolvent behaviour associat…
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Using their method, Rigger proved the same theorem for Riemannian manifolds being asymptotically equal to the (spatial) (Schwarzschild-)Anti-de Sitter solution. This was generalized to asymptotically hyperbolic manifolds by Neves-Tian, Chodosh, and the author at a later stage. In this work, we prove the reverse implication as the author already did in …
We revisit Vasy's method for showing meromorphy of the resolvent for (even) asymptotically hyperbolic manifolds. It provides an effective definition of resonances in that setting by identifying them with poles of inverses of a family of Fredholm differential operators. In the Euclidean case the method of complex scaling made this available since the 70's but in the hyperbolic case an effective definition was not known until recently. Here we present a simplified version which relies only on standard pseudodifferential techniques and estimates for hyperbolic operators. As a byproduct we obtain …
We verify the 3-dimensional Glassey conjecture on asymptotically flat manifolds $(R^{1+3}, g)$, where the metric $g$ is certain small space-time perturbation of the flat metric, as well as the nontrapping asymptotically Euclidean manifolds. Moreover, for radial asymptotically flat manifolds $(R^{1+n}, g)$ with $n\ge 3$, we verify the Glassey conjecture in the radial case. High dimensional wave equations with higher regularity are also discussed. The main idea is to exploit local energy and KSS estimates with variable coefficients, together with the weighted Sobolev estimates including trace es…
By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angular momentum) black hole backgrounds, long range asymptotically Euclidean spaces, and small time-dependent asymptotically flat perturbations of Minkowski space-time. We also permit lower order perturbations of the wave operator. The key estimates are a class of weighted Strichartz …
In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem.
We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to $\mathbb{R}^n$.
We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper \cite{km} have important role in this paper.dan curves and is extended to the case of $C^{2,α}$ surfaces with smooth and compact boundary.
We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low frequencies. We show in particular that the damped wave behaves like a solution of a heat equation which depends on the H-limit of the metric and the mean value of the absorption index.
We prove families of uniform $(L^r,L^s)$ resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author \cite{KRS}. In the case of the sphere we take advantage of the fact that the half-wave group of the natural shifted Laplacian is periodic. In the case of hyperbolic space, the key ingredient is a natural variant of the Stein-Tomas restriction theorem.
This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain a few new results concerning the $n,1$ relative and global capacities. We obtain sharp estimates for the $n,1$ relative capacity of the concentric condensers $(\overline{B}(0,r), B(0,1))$ for all $r$ in $[0,1).$ As a consequence we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R…
In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds, where such a decay rate has been conjectured by R. Price. Our results apply to both of these cases.
We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds, in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajlasz-Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.
We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying suff…
