Noel Brady, Benson Farb
In this paper we construct and study isoperimetric functions at infinity for Hadamard manifolds. These quasi-isometry invariants give a measure of the spread of geodesics in such a manifold.
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In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes from a previous result of P. Chruściel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics.
For a manifold with nonpositive curvature, the Martin boundary is described by the behavior of normalized Green's functions at infinity. A classical result by Anderson and Schoen states that if the manifold has pinched negative curvature, the geometric boundary is the same as the Martin boundary. In this paper, we study the Martin boundary for rank 1 manifolds admitting compact quotients. It is proved that a residual set in the geometric boundary can be identified naturally with a subset of the Martin boundary. This gives a partial answer to one of the open problems in geometry collected by Ya…
We study the filling invariants at infinity div_{k} for Hadamard manifolds defined by Brady and Farb in [Filling-invariants at infinity for manifolds of nonpositive curvature]. Among other results, we give a positive answer to the question they posed: whether these invariants can be used to detect the rank of a symmetric space of noncompact type.
We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval $[(n-1)^2K/4, \infty)$ if an $n$-dimensional Riemannian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant $(n-1)K$ converges to zero at infinity.
We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequence, we obtain that the fundamental group of M cannot be isomorphic to the fundamental group of any Riemannian manifold of nonpositive sectional curvature. In particular, M is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvatu…
In this paper we study a homological version of the higher-dimensional divergence invariants defined by Brady and Farb. We show that they are quasi-isometry invariants in the class of proper cocompact Hadamard spaces in the sense of Alexandrov and that they can moreover be used to detect the Euclidean rank of such spaces. We thereby extend results of Brady-Farb, Leuzinger, and Hindawi from the setting of symmetric spaces of non-compact type to that of singular Hadamard spaces. Finally, we exhibit the optimal power for the growth of the divergence above the rank for symmetric spaces of non-comp…
We define an invariant for compact spin manifolds $X$ of dimension $4k$ equipped with a metric $h$ of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of $h$ to be the conformal infinity of a conformally compact Einstein metric on $X$.
In this article, we extend Anderson's higher-dimensional Dehn filling construction to a large class of infinite-volume hyperbolic manifolds. This gives an infinite family of topologically distinct asymptotically hyperbolic Einstein manifolds with the same conformal infinity. The construction involves finding a sequence of approximate solutions to the Einstein equations and then perturbing them to exact ones.
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
Measured foliations at infinity of quasi-Fuchsian manifolds
本文研究拟Fuchsian流形无穷远边界上的可测叶状结构对，证明了接近Fuchsian情形时该叶状结构对具有填充性，并进一步证明任意填充对在充分小的尺度下均可实现为某拟Fuchsian流形的无穷远叶状结构，从而回答了Schlenker在Fuchsian轨迹附近提出的问题。
On the problem of filling by a Poincaré-Einstein metric in dimension 4
本文研究四维空间中通过庞加莱-爱因斯坦度量填充问题，建立紧致性结果并推导填充存在性结论。
On the Poincaré-Einstein manifolds with cylindrical conformal infinity
本文证明具有柱形共形无穷远的非正曲率Poincaré-Einstein流形必为双曲，并在圆因子长度足够大或Weyl曲率能量足够小时给出唯一性，通过建立新的ε-正则性实现。
This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability in this class depends on the topology of the filling manifold. The obstruction to solvability for arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions.
Strichartz and Spectral Projection Estimates on Asymptotically Conic Manifolds
论文在渐近锥流形上证明了无损失单位区间Strichartz定理和谱投影定理，研究了曲率和局部平滑估计对结果的影响。
Any one-cusped hyperbolic manifold M with an unknotting tunnel tau is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that tau is isotopic to a geodesic, and characterize whether tau is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of tau relative to a maximal cusp. These results give generic answers to three long-standing questions posed by Adams, Sakuma, and Weeks. We also construct an explicit sequence of one-tunnel knot…
A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds
研究关于庞加莱 - 爱因斯坦流形共形填充的不等式，通过特定方法证明了精确相对比较不等式并建立刚性，证实了孙咏怡·张的猜想。
Nearly geodesic surfaces are filling
研究证明了近测地面和高 genus 测地面在闭合双曲3流形中是填充的，且给出了填充面的下界和间隙定理。
Convexity on manifolds without focal points and applications
本文研究了无焦点点的黎曼流形上的严格凸函数，通过几何定义的凸函数推导出如等周轮廓函数连续性等结论，并在Kähler流形上获得Stein性质和度量球体积增长的下界。主要应用涉及拉普拉斯算子的谱，证明了在径向曲率非正的条件下，绝对连续部分谱包含无限区间，并利用无穷远处的几何构造严格凸函数，最终证明了Cheeger常数与体积熵的相等性。
Complete 3-manifolds of positive scalar curvature with quadratic decay
本文证明具有正数量曲率且无穷远处次二次衰减的完备可定向3-流形可分解为球面流形与S²×S¹的连通和，推广了Gromov-Wang定理并部分回答了Gromov猜想，且该衰减速率是最优的。
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