Felix Finster, Ines Kath
We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the total mass tends to zero.
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In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact subset are unique. Therefore we are able to conclude that there is a unique foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass.
On conical asymptotically flat manifolds
本文证明了Petrunin和Tuschmann关于不存在渐近于半平面的渐近平坦4-流形的猜想，并综述了作为研究动机的引力瞬子相关进展与问题。
We prove existence and uniqueness of foliations by stable spheres with constant mean curvature for 3-manifolds which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass. These metrics arise naturally as spacelike timeslices for solutions of the Einstein equation with a negative cosmological constant.
We consider an asymptotically flat Lorentzian manifold of dimension (1,3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the ADM energy tends to zero.
In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymptotically flat manifolds. In the case of dimension $n=3$ or 4, we obtain that the ADM mass is invariant under the Yamabe flow and the Yamabe flow is the gradient flow of Einstein-Hilbert functional on asymptotically flat manifolds
In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $\ge 6$. The purpose of the present paper is to use a different way to exhibit a family of complete $I$-dimensinal ($I\ge5$) Riemannian manifolds of positive Ricci curvature, quadratically asymptotically nonnegative sectional curvature, and certain infinite Betti number $b_j$ ($2\le j\le I-2$).
Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets
本文构造了接近反德西特-史瓦西双曲面的初始数据集的常时空平均曲率曲面族，并应用于定义渐近双曲初始数据集的质心。
In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive mass, the foliation of stable spheres of constant mean curvature is unique.
In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generize it to the manifolds with $k$-th asymptotically nonnegative Ricci curvature by using extensions of Abresch-Gromoll's excess function estimate.
We study Ricci flows on $R^n$, $n\ge 3$, that evolve from asymptotically flat initial data. Under mild conditions on the initial data, we show that the flow exists and remains asymptotically flat for an interval of time. The mass is constant in time along the flow. We then specialize to the case of rotationally symmetric, asymptotically flat initial data containing no embedded minimal hyperspheres. We show that in this case the flow is immortal, remains asymptotically flat, never develops a minimal hypersphere, and converges to flat Euclidean space as the time diverges to infinity. We discuss …
In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature.
Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds
本文证明在渐近平坦（标量曲率非负）和渐近双曲（标量曲率≥-6）流形中，对任意正常数c（分别c>0和c>2），存在紧致、几乎嵌入、自由边界的常平均曲率曲面，证明结合min-max理论与逆平均曲率流的长期光滑性。
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.
Let $(M, g)$ be an asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature and positive mass. We show that each leaf of the canonical foliation through stable constant mean curvature surfaces of the end of $(M, g)$ is uniquely isoperimetric for the volume it encloses.
In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n under some conditions on the density of rays starting from the base point p or on the volume growth of geodesic balls in M.
Sobolev Inequality on Manifolds With Asymptotically Nonnegative Bakry-Émery Ricci Curvature
本文受已有工作启发，在具有密度且Bakry-Émery Ricci曲率渐近非负的流形上证明了Sobolev不等式。
Using the ABP-method as in a recent work by Brendle, we establish some sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds in a complete Riemannian manifold with asymptotically nonnegative curvature. These inequalities generalize those given by Brendle in the case of complete Riemannian manifolds with nonnegative curvature.
We show that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature is isometric to flat $\mathbb{R}^3$ if it admits an unbounded area-minimizing surface. This answers a question of R. Schoen.
A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Qing}.
Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cubic volume growth are ALF.
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.
Let $(M^3, g)$ be an asymptotically flat 3-manifold with positive ADM mass. In this paper, we show that each leaf of the canonical foliation is the unique isoperimetric surface for the volume it encloses. Our proof is based on the "fill-in" argument and sharp isoperimetric inequality on asymptotically flat 3-manifold with nonnegative scalar curvature.
Let $(M,g)$ be an asymptotically flat $3$-manifold containing no closed embedded minimal surfaces. We prove that for every point $p\in M$ there exists a complete properly embedded minimal plane in $M$ containing $p$.
We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with $g$ on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: Given a function $f$ on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equi…
We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any `small' asymptotically cylindrical Ricci-flat deformations of asymptotically cylindrical Ricci-flat Kähler metrics are again Kähler, possibly with respect to a perturbed complex structure. We also find the dimension of the moduli space for these small deformations. In the class o…
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 $S^2$.
By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected $C^2$-smooth surface as initial data in asymptotically ADS-Schwarzschild manifolds…
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 $\mathbb{R}^n$.
Curvature pinching of asymptotically conical gradient expanding Ricci solitons
本文研究了完整非紧致渐近锥形梯度扩展里奇溶剂的曲率捏紧现象，建立了若干Hamilton-Ivey型曲率捏紧估计，包括三维梯度里奇扩展体的正截面曲率结果。
A sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature
本文利用黎曼几何标准比较方法，在渐近非负曲率的完备非紧流形中为闭超曲面建立了锐利几何不等式，并简化了非负里奇曲率情形下Willmore型不等式的证明。
We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.
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.
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 paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,α}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order near boundary have conformal compactifications that are $C^{m+2,α}$ up to the boundary when Weyl curvature is in $C^{m,α}$ and the boundary metric is in $C^{m+2,α}$ where $m\geq 3.$
Monotonicity of harmonic functions on $3$-manifolds with an asymptotically flat end
研究了三维流形上具有渐近平坦端的正调和函数的单调性，并探讨了具有两个端的Schwarzschild流形的刚性特征。
Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow
针对平均曲率流中的平均凸渐近锥形自膨胀子，论文证明了当 $n\geq 3$ 且渐近锥平均凸、弱凸时，该完备自膨胀子必为严格凸。
We study the Yamabe flow starting from an asymptotically flat manifold $(M^n,g_0)$. We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if $Y(M,[g_0])>0$, and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.
We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain sense.
A positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary
本文在带非紧边界的光滑渐近平滑黎曼流形上，证明了在紧超曲面处仅连续的度量的正质量定理，并将其应用于带非紧角流形的情形。
We follow the approach employed by Y. Choquet-Bruhat, J. Isenberg and D. Pollack in the case of closed manifolds and establish existence and non-existence results for the Einstein-scalar field constraint equations on asymptotically hyperbolic manifolds.
In this paper we prove a rigidity result for the equality case of the Penrose inequality on $3$-dimensional asymptotically flat manifolds with nonnegative scalar curvature and corners. Our result also has deep connections with the equality cases of Theorem 1 in \cite{Miao2} and Theorem 1.1 in \cite{LM}.
We prove that Maxwell fields of asymptotically flat solutions of the Einstein-Maxwell equations inherit the stationarity of the metric.
In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds with curvature decay and controlled holonomy. As an application, we show that any complete asymptotically flat Ricci-flat metric on a $4$-manifold which is homeomorphic to $\mathbb R^4$ must be isometric to the Euclidean or the Taub-NUT metric, provided that the tangent cone at infinity is not $\mathbb R \times \mathbb R_+$.
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of visual and harmonic measures on the ideal boundary. Then, we prove an asymptotic analogue of the classical mean value property of harmonic manifolds, and we characterize asymptotically harmonic manifolds, among Cartan-Hadamard spaces of strictly negative curvature, by the existen…
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant cones are rotationally symmetric.
This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold $(M,g)$ of real dimension $4n+2$ which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. The fact that we do not assume $g$ to be Kähler reflects in the inequality for the scalar curvature.
Adapting Israel's proof of static black hole uniqueness, we show that the Schwarzschild spacetime is the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere.
We consider the question whether a static potential on an asymptotically flat 3-manifold can have nonempty zero set which extends to the infinity. We prove that this does not occur if the metric is asymptotically Schwarzschild with nonzero mass. If the asymptotic assumption is relaxed to the usual assumption under which the total mass is defined, we prove that the static potential is unique up to scaling unless the manifold is flat. We also provide some discussion concerning the rigidity of complete asymptotically flat 3-manifolds without boundary that admit a static potential.
We prove that any asymptotically Euclidean metric on $\mathbb{R}^n$ with no conjugate points must be isometric to the Euclidean metric.
We show an improvement of Bray sharp mass-capacity inequality and Bray-Miao sharp upper bound of the capacity of the boundary in terms of its area, for three-dimensional, complete, one-ended asymptotically flat manifolds with compact, connected boundary and with nonnegative scalar curvature, under appropriate assumptions on the topology and on the mean curvature of the boundary. Our arguments relies on two monotonicity formulas holding along level sets of a suitable harmonic potential, associated to the boundary of the manifold. This work is an expansion of the results contained in the PhD the…
We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold for more general classes of asymptotically flat data.
Scalar curvature lower bounds on asymptotically flat manifolds
本文研究渐近平坦流形上分布意义与β-弱意义的标量曲率下界，证明Ricci-DeTurck流下的标量曲率下界与β-弱下界及时间相关，且W^{1,p}度量的分布标量曲率下界与无穷远β-弱下界一致。
Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in $c_1(L)$ and its automorphism group Aut$(M,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case when Aut$(V,L)$ is not discrete.
Intrinsic flat stability of the positive mass theorem for asymptotically hyperbolic graphical manifolds
本文研究渐近双曲图形流形正质量定理的稳定性，借鉴Huang、Lee和Sormani的方法，证明当总质量几乎为零时该类流形在内蕴平坦距离意义下接近双曲空间。
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
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 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.
On the uniqueness of surfaces of constant spacetime mean curvature in asymptotically Schwarzschildean lightcones
研究渐近平直史瓦西光锥中常时空平均曲率曲面的唯一性，证明对于一般渐近平直概念存在唯一渐近平直叶状结构，此前已证存在性，此次在更弱假设下证唯一性。
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivity of the isoperimetric mass established by G. Huisken we prove an existence result for large isoperimetric regions in general asymptotically flat three manifolds with non-negative scalar curvature.
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