Leonardo Marazzi
We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary, and also prove that there is always an essential singularity of the resolvent in this setting. We use this analysis to prove an inverse result for conformally compact odd dimensional Einstein manifolds.
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We prove propagation of weighted Sobolev regularity for solutions of the hyperboloidal Cauchy problem for a class of quasi-linear symmetric hyperbolic systems, under structure conditions compatible with the Einstein-Maxwell equations in space-time dimensions $n+1\ge 7$. Similarly we prove propagation of polyhomogeneity in dimensions $n+1\ge 9$. As a byproduct we obtain, in those last dimensions, polyhomogeneity at null infinity of small data solutions of vacuum Einstein, or Einstein-Maxwell equations evolving out of initial data which are stationary outside of a ball.
An Isomorphism Theorem for Some Linear Elliptic Differential Operators on Quasi-Asymptotically Conical Manifolds
该研究针对拟渐近锥形流形，探究了一类带无界$b$-向量场的线性椭圆微分算子，为相关几何分析问题提供算子层面的理论支撑。
Asymptotics for resolutions and smoothings of Calabi-Yau conifolds
研究通过多项式同质展开分析Calabi-Yau锥形的解析和光滑退化行为，利用吹胀和粘合技术构造多项式同质Kähler度量族。
We define scattering phases associated to pairs of Laplacians on asymptotically hyperbolic manifolds, and prove some spectral asymptotics for them. These result are applications of Isozaki-Kitada's constructions which we adapt to this framework.
We prove high energy estimates for the boundary values of the weighted resolvent of the Laplacian on an asymptotically hyperbolic manifold. Our point is to use weights that fit the pseudo-differential calculus associated with the asymptotically hyperbolic geometry. Our proof is a combination of a trick using Mourre theory and of results by Froese-Hislop and Cardoso-Vodev.
Resolvent estimate for asymptotically hyperbolic surfaces without evenness assumption
本文证明了一类非紧流形上拉普拉斯算子的尖锐截断预解估计，并将其应用于负曲率渐近双曲曲面，得到无需无穷远偶性假设的全局Strichartz与谱投影估计。
Upper bound on the number of resonances for even asymptotically hyperbolic manifolds with real-analytic ends
本文证明了具有实解析端的偶渐近双曲流形在半径趋于无穷大的圆盘内共振数量的多项式上界，并为Schwarzschild-de Sitter时空的拟正规频率数量提供了类似上界。
We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the compactification. We also show that these Calabi-Yau metrics admit a polyhomogeneous expansion at infinity, a result that we extend to asymptotically conical Calabi-Yau metrics as well. We then study the moduli space of Calabi-Yau deformations that fix the complex structure at infinity. There is a Weil-Petersson metric on this space which we show is Kähler. By p…
The wave kernel on asymptotically complex hyperbolic manifolds
该研究针对渐近复双曲流形上的拉普拉斯波核，证明其属于特定傅里叶积分算子类，分析其迹的奇异性分布并得到零时刻渐近展开，推进了相关流形上的谱几何研究。
In analogy with radiation fields for asymptotically Euclidean manifolds, introduced by F.G. Friedlander, we define radiation fields for asymptotically hyperbolic manifolds. We use them to give a translation representation of the wave group and to obtain the scattering matrix for such manifolds. Furthermore, we prove a support theorem for the radiation fields which generalizes to this setting the well known support theorem of Helgason and Lax and Phillips for the horocyclic Radon transform. This is used to show that the manifold and the metric, up to invariants, are determined by the scattering…
We prove a local well-posedness theorem for the (n+1)-dimensional Einstein equations in Lorentzian signature, with initial data $(\tilde g, K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\hat g$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an n-dimensional asymptotically hyperbolic Riemannian manifold $(M,\tilde g)$ such that the conformally rescaled metric $x^2 \tilde g$ (with $x$ a boundary defining function) extends to the closure $\bar M$ of $M$ as a metric of class $C^{n-1}…
For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove a fractal upper bound on the number of resonances near the essential spectrum, with power determined by the dimension of the trapped set. This covers the case of general convex cocompact quotients (including the case of connected trapped sets) where our result implies a bound on the number of zeros of the Selberg zeta function in disks of arbitrary size along the imaginary axis. Although no sharp fractal lower bounds are known, the case of quasifuchsian groups, included here, is most likely to provide them.
The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds
本文研究渐近双曲流形上薛定谔算子负特征值的计数函数渐近性，给出势条件判定有限或无限个负特征值，并在无限情形下得到区间$(-\infty,-E)$内特征值计数当$E\to 0$时的渐近行为。
Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, $g.$ A model form is established for such metrics close to the boundary. It is shown that the scattering matrix at energy $ζ$ exists and is a pseudo-differential operator of order $2ζ+1 - \dim X.$ The symbol of the scattering matrix is then used to show that except for a countable set of energies the scattering matrix at one energy determines the diffeomorphism class of the metric modulo terms vanishing to infinite order at the boundary. An analogous result is proved for potent…
We study the scattering relation and the sojourn times on non-trapping asymptotically hyperbolic manifolds and use it to obtain the asymptotics of the distance function on geodesically convex asymptotically hyperbolic manifolds.
We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schrödinger operators $P=Δ+V$ with short range potential $V$ on asymptotically conic manifolds $(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$. We make the assumption that the dimension is greater or equal to 3 and that $P$ has no $L^2$ null space and no resonance at 0. In particular, we show that the Schwartz kernel of $R(k)$ is a conormal polyhomogeneous distribution on a desingularized version of $M\times M\times [0,1]$. Using this, we show that the Riesz transform of $P$ is bounded on $L^p$ for $1<p<n$ and that t…
As a consequence of a result of Cardoso and Vodev, we show that the resolvent of the Laplacian on asymptotically hyperbolic manifolds is analytic in an exponential neighbourhood of the critical line. The case of non-trapping metrics with constant curvature near infinity is also considered: there exists a strip with at most a finite number of resonances.
We describe how the global operator induced on the boundary of an asymptotically Minkowski space links two asymptotically hyperbolic spaces and an asymptotically de Sitter space, and compute the scattering operator of the linked problem in terms of the scattering operator of the constituent pieces.
We prove that the scattering matrix at all energies restricted to an open subset of the boundary determines an asymptotically hyperbolic manifold modulo isometries that are equal to the identity on the open subset where the scattering matrix is known.
We construct a semi-classical parametrix for the Laplacian on non-trapping asymptotically hyperbolic manifolds, which generalizes the construction of Melrose, Sa Barreto and Vasy. As applications, we obtain high energy resolvent estimates and resonance free strips. Also, we prove that the Friedlander radiation field decays exponentially for initial data in proper spaces.
We show that resonant states in scattering on asymptotically hyperbolic man-ifolds that are analytic near conformal infinity, have analytic radiation patterns at infinity. On even asymptotically hyperbolic manifolds we also show that smooth solutions of Vasy operators with analytic coefficients are also analytic. That answer a question of M.Zworski ([14] Conjecture 2). The proof is based on previous results of Baouendi-Goulaouic and Bolley-Camus-Hanouzet and for convenience of the reader we present an outline of the proof of the latter.
We consider an inverse problem associated with $n$-dimensional asymptotically hyperbolic orbifolds $(n \geq 2)$ having a finite number of cusps and regular ends. By observing solutions of the Helmholtz equation at the cusp, we introduce a generalized $S$-matrix, and then show that it determines the manifolds with its Riemannian metric and the orbifold structure.
We show that the resolvent of the Laplacian on asymptotically hyperbolic spaces extends meromorphically with finite rank poles to the complex plane if and only if the metric is `even' (in a sense). If it is not even, there exist some cases where the resolvent has an essential singularity in the non-physical sheet.
We obtain a formula for the Schwartz kernel of the scattering operator in terms of the Schwartz kernel of the fundamental solution of the wave operator on asymptotically hyperbolic manifolds. If there are no trapped geodesics, this formula is used to show that the scattering operator is a Fourier integral operator that quantizes the scattering relation.
We prove local in time Strichartz estimates without loss for the restriction of the solution of the Schroedinger equation, outside a large compact set, on a class of asymptotically hyperbolic manifolds.
In this paper we construct a parametrix for the high-energy asymptotics of the analytic continuation of the resolvent on a Riemannian manifold which is a small perturbation of the Poincaré metric on hyperbolic space. As a result, we obtain non-trapping high energy estimates for this analytic continuation.
This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we use semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, along with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on general non- trapping asymptotically hyperbolic manifolds, generalizing the work due to Melrose, Sa Barreto and Vasy. We note that there is an 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 provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a problem left open in \cite{SSWZ} about the endpoint case for global well-posedness of nonlinear wave equations. We also provide estimates in this context for the maximal wave operator.
We study the high energy estimate for the resolvent $R(λ)$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHM). In the literature, polynomial bound of the form $\|R(λ)\| = O(|λ|^{N})$ for $|λ|$ large and $λ\in \mathbb{C}$ in strips where $R(λ)$ is holomorphic was established for some $N > -1$. We prove the optimal bound $O(|λ|^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.
On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calderón problem for the Conformal Laplacian
本文通过渐近双曲流形与Conformal Laplacian的关系，证明了特定能量下的散射矩阵能确定边界度量的jet结构。
This paper is concerned with an abstract dissipative hyperbolic equation with time-dependent coefficient. Under an assumption which ensures that the energy does not decay, this paper provides a condition on the coefficient, which is necessary and sufficient so that the solutions tend to the solutions of the free wave equation.
Applying the spectral measure estimates obtained in the author's joint work with A. Hassell, we establish global-in-time Strichartz estimates without loss via truncated / microlocalized dispersive estimates as well as energy estimates.
We study the normal operator to the geodesic X-ray transform on functions in the setting of simple asymptotically hyperbolic manifolds. We construct a parametrix for the normal operator in the 0-pseudodifferential calculus and use it show a stability estimate.
We solve the Jang equation with respect to asymptotically hyperbolic "hyperboloidal" initial data. The results are applied to give a non-spinor proof of the positive mass theorem in the asymptotically hyperbolic setting. This work focuses on the case when the spatial dimension is equal to three.
We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from \cite{CH2} and arguments borrowed from \cite{HZ, Zhang}. As an application, we prove the small data global existence for any power $p\in(1, 1+\frac{4}{n-1})$ for the shifted wave equation in this setting, involving nonlinearities of the form $\pm|u|^p$ or $\pm|u|^{p-1}u$, which answers partially an open question raised in \cite{SSW}.
We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi--Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on $S^2$.
We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recently, roughly by extending the problem across the boundary of the compactification of the asymptotically hyperbolic space in a suitable manner. The main novelty is that the non-scalar nature of the operator is dealt with by relating it to a problem on an asymptotically Minkowski spa…
In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
Mean first escape times of Brownian motion on asymptotically hyperbolic and gas giant metric surfaces
研究了布朗运动在渐近双曲和气态巨行星度量表面的平均首次逃逸时间，通过渐近展开和数值模拟验证了不同度量下逃逸时间的行为差异。
In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a theorem due to De Lellis and Müller.
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 …
On Jang's equation and the Positive Mass Theorem for asymptotically hyperbolic initial data sets with dimensions above three and below eight
本文在维度4至7的渐近双曲初始数据上求解Jang方程，为非旋量证明正质量定理提供了新途径，并推广了Sak21在三维情形下的结果。
Upper and lower bounds on the heat kernel on complete Riemannian manifolds were obtained in a series of pioneering works due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. However, these estimates do not give a complete picture of the heat kernel for all times and all pairs of points. Inspired by the work of Davies-Mandouvalos on $\mathbb{H}^{n + 1}$, we study heat kernel bounds on Cartan-Hadamard manifolds that are asymptotically hyperbolic in the sense of Mazzeo-Melrose. Under the assumption of no eigenvalues and no resonance at the bottom of the continuous spectrum, we show that the heat kernel o…
Decay properties of Vlasov fields on non-trapping asymptotically hyperbolic manifolds
本文研究非捕获渐近双曲流形上Vlasov场的时间衰减，证明二维情形下空间密度的最优衰减估计：远离零速度集时指数衰减，一般情形逆多项式衰减，方法为交换向量场。
We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity, and introduce several technical tools for studying PDEs on such manifolds. Our results employ two novel families of function spaces suitable for metrics potentially having a large amount of interior differentiability, but whose Sobolev regularity implies only a Hölder continuous conformal structure at infinity. We establish Fredholm theorems for elliptic operators arising from metrics in these families. To demonstrate the utility of our methods, we solve the Yamabe problem in this category. As a specia…
Sectoriality of the Laplacian on Asymptotically Hyperbolic Spaces
本文证明了渐近双曲度量下函数拉普拉斯算子与对称2-张量Lichnerowicz拉普拉斯算子在加权Hölder空间中的扇形性，并由此利用解析半群理论得到抛物型发展方程的适定性。
On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric tensors, a similar result is proven for (convex cocompact) quotients of hyperbolic space.
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
A Bahri-Brezis type problem on asymptotically hyperbolic manifolds
该研究聚焦紧致三维渐近双曲流形上的Bahri-Brezis型问题，采用Bahri-Coron代数拓扑方法，在特定退化边值问题的正性条件下证明了解的存在性。
We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite -- an altered version of -- the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic co…
In this paper, we study some intrinsic characterization of conformally compact manifolds. We show that, if a complete Riemannian manifold admits an essential set and its curvature tends to -1 at infinity in certain rate, then it is conformally compactifiable and the compactified metrics can enjoy some regularity at infinity. As consequences we prove some rigidity theorems for complete manifolds whose curvature tends to the hyperbolic one in a rate greater than 2.
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this CMC-foliation. Rigger and Neves-Tian showed in 2004 and 2009/10 analogous existence and uniqueness theorems for three-dimensional, asymptotically Anti-de Sitter and asymptotically hyperbolic manifolds with positive mass aspect function, respectively. Last year, Cederbaum-Cortier-Sakovich proved that the CMC-foliation characterizes the center of mass in the hyperbolic s…
In this paper we pursue the work initiated in \cite{Bahuaud, BahuaudGicquaud}: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to -1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tenso…
We consider the Laplacian $Δ$ on an asymptotically hyperbolic manifold $X$, as defined by Mazzeo and Melrose. We give pointwise bounds on the spectral measure for the operator $(Δ- n^2/4)_+^{1/2}$ on such manifolds, under the assumptions that $X$ is nontrapping and there is no resonance at the bottom of the spectrum. This uses the construction of the resolvent given by Mazzeo and Melrose (valid when the spectral parameter lies in a compact set), Melrose, Sá Barreto and Vasy (high energy estimates for a perturbation of the hyperbolic metric) and the present authors (see also the work by Yiran W…
We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius $α$. Static solutions in this model are shown to exhibit an interesting bifurcation pattern in the parameter $α$. We relate this pattern to the Morse index of the static solution with maximal energy. Using a hyperboloidal approach to the initial value problem, we describe the relaxation to the ground state solution for generic initial data and unstable static solutions for initial data of codimension one,…
In this paper, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. We define a mass term of the AH end. If the $σ_{2}$ curvature has lower bound $σ_{2}\geq\frac{3}{2}$, we prove a Penrose type inequality relating the mass and contributions from singularities. We also classify sharp cases, which is the standard hyperbolic 4-space $\mathbb{H}^{4}$ when no singularity occurs. It is worth noting that our curvature condition implies non-positive energy density.
A definition of the mass aspect function for weakly regular asymptotically hyperbolic manifolds
本文针对弱正则渐近双曲流形，提出了一种ADM风格的质量方面函数定义，统一了Wang和Chruściel-Herzlich的两种质量定义方法，并证明了其可通过Ricci张量计算及在无穷远图表变换下的良好协变性质。
We prove microlocal estimates at the trapped set of asymptotically Kerr spacetimes: these are spacetimes whose metrics decay inverse polynomially in time to a stationary subextremal Kerr metric. This combines two independent results. The first one is purely dynamical: we show that the stable and unstable manifolds of a decaying perturbation of a time-translation-invariant dynamical system with normally hyperbolic trapping are smooth and decay to their stationary counterparts. The second, independent, result provides microlocal estimates for operators whose null-bicharacteristic flow has a norm…
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