Mohamad A. Hindawi
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.
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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…
After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we discuss what is known and the difference between the two-dimensional and higher-dimensional cases. Turning our attention to the two-dimensional case, we prove that the Dirichlet problem at infinity on a two-dimensional Cartan-Hadamard manifold is solvable under the curvature condition $K\leq (1+ε)/(r^2 \log r)$, outside of a compact set, for some $ε>0$ in polar coordinates around some pole. This condition on the curvature is sharp, and improves upon the previously known case of quadratic curvature decay. Finall…
We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.
It is well known that simply connected symmetric spaces of non-positive sectional curvature admit a linear isoperimetric filling inequality for cycles of dimension greater than or equal to the rank of the space. In this note we extend that result to homogeneous Hadamard manifolds.
We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to -infinity and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to -infinity faster than any…
Let $M$ be Hadamard manifold with sectional curvature $K_{M}\leq-k^{2}$, $k>0$. Denote by $\partial_{\infty}M$ the asymptotic boundary of $M$. We say that $M$ satisfies the strict convexity condition (SC condition) if, given $x\in\partial_{\infty}M$ and a relatively open subset $W\subset\partial_{\infty}M$ containing $x$, there exists a $C^{2}$ open subset $Ω\subset M$ such that $x\in\operatorname*{Int}(\partial_{\infty}Ω) \subset W$ and $M\setminusΩ$ is convex. We prove that the SC condition implies that $M$ is regular at infinity relative to the operator $$\mathcal{Q}[u] :=\mathrm{div}(\frac…
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.
In a previous paper, we proved that a quasi-isometric map $f:X\longrightarrow Y$ between two pinched Hadamard manifolds $X$ and $Y$ is within bounded distance from a unique harmonic map. We extend this result to maps $f:Γ\backslash X\longrightarrow Y$, where $Γ$ is a convex cocompact discrete group of isometries of $X$ and $f$ is locally quasi-isometric at infinity.
Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds
研究了Cartan-Hadamard流形中带有渐近Dirichlet条件的Killing图的均曲率演化先验估计，证明了相应抛物型流的存在性，并通过构造合适的障碍来保证解的正则性。
We investigate the geometry and topology of extremal domains in a manifold with negative sectional curvature. An extremal domain is a domain that supports a positive solution to an overdetermined elliptic problem (OEP for short). We consider two types of OEPs. First, we study narrow properties of such domains in a Hadamard manifold and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity and how the domain behaves at infinity. This shows interesting relations with the Singular Yamabe Problem. Later, we focus on extremal domains i…
We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity. This work extends the results of Lehtonen (2016) to dimensions $n \geq 3$ and to the case of tensor fields of any order.
Given an unbounded domain $Ω$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the sectional curvature of $M$ is $\le -1$ this Dirichlet problem is solvable if $Ω$ satisfies certain convexity condition at infinity and if $\partial Ω$ is mean convex. We also prove that mean convexity of $\partial Ω$ is a necessary condition, extending to unbounded domains some r…
Translating solitons over Cartan-Hadamard manifolds
本文证明了 Cartan-Hadamard 流形上平均曲率流图形平移孤立子的存在性，揭示了其渐近行为对流形曲率的依赖性，并在特定曲率条件下探讨了有界解及渐近 Dirichlet 问题的可解性。
We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian geometry. We also estimate the volume growth entropy for the balls in such manifolds.
We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the topology at infinity of $f$-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted $L^1$-Sobolev inequality for hypersurfaces in Cartan-Hadamard weighted manifolds, satisfying suitable restrictions on the weight fun…
It is proved the existence of entire solutions of the Laplace's and minimal hypersurface's PDEs on a Hadamard manifold $M$ under certain curvature conditions by investigating the asymptotic Dirichlet's problems for these PDEs. In the harmonic case it is obtained an existence result which assumes the same growth condition on the sectional curvature as of Theorem 1.2 of E. Hsu \cite{Hsu} but that contemplates cases having Ricci curvature with exponential decay. It is also obtained a result which extends and improves Theorem 3.6 of Choi \cite{Choi}. In the minimal case one obtains an extension an…
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.
Ground states for aggregation-diffusion models on Cartan-Hadamard manifolds
本文研究Cartan-Hadamard流形上聚集-扩散模型自由能泛函的基态存在性，给出了截面曲率有界条件下基态存在的充要条件，并揭示了吸引势增长阶数对扩散行为的控制作用。
The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact manifold with Ballmann rank one, we show that Martin points are generic and of full harmonic measure. The result of this paper provides a partial answer to an open problem of S. T. Yau.
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.
In this paper we consider Cartan-Hadamard manifolds (i.e. simply connected of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold when the curvature is bounded, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Caldéron-Zygmund inequalities and the $L^p$-positivity preserving property, i.e. $u\in L^p\ \&\ (-Δ+ 1)u\ge 0 \Rightarrow u\ge 0$. The main tool is a new class of first and second order Hardy-type inequalities on Ca…
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。
Let $M$ be a Cartan-Hadamard manifold with sectional curvature satisfying $-b^2\leq K\leq -a^2<0$, $b\geq a>0.$ Denote by $\partial_{\infty}M$ the asymptotic boundary of $M$ and by $\bar M:= M\cup\partial_\infty M$ the geometric compactification of $M$ with the cone topology. We investigate here the following question: Given a finite number of points $p_{1},...,p_{k}\in\partial_\infty M,$ if $u\in C^{\infty}(M)\cap C^{0}\left( \bar{M}\backslash\left\{ p_{1},...,p_{k}\right\} \right) $ satisfies a PDE $\mathcal Q(u)=0$ in $M$ and if $u|_{\partial_\infty M\setminus\left\{p_{1},...,p_{k}\right\}}…
The Lane-Emden system on Cartan-Hadamard manifolds: asymptotics and rigidity of radial solutions
本文研究了Cartan-Hadamard流形上Lane-Emden系统正径向解的存在性、存在区域维度、渐近行为二分性及有限能量解的刚性结果。
Estimates of the Poisson kernel on negatively curved Hadamard manifolds
本文研究夹挤负曲率 Hadamard 流形的 Poisson 核，借助 Anderson-Schoen 的正调和函数锥估计证明其整体上下界，并用于量化调和测度向边界点测度及无穷远边界测度的收敛。
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.
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…
Gradient descent for unbounded convex functions on Hadamard manifolds and its applications to scaling problems
研究Hadamard流形上无界凸函数的梯度流渐进行为，建立梯度范数与退化函数的对偶定理，并应用于非交换优化和算子缩放问题，揭示梯度下降收敛性质及矩阵 pencil 的克罗内克标准型。
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。
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