Gérard Besson, Gilles Courtois, Sylvain Gallot
Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $Γ$ acting on $X$, then either $Γ$ is virtually nilpotent or the algebraic entropy $Ent (Γ) \geq C(n,a)$.
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We prove dynamical stability of a natural class of hypersurface laminations defined over Cartan--Hadamard manifolds of pinched curvature. We achieve this by providing a complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds, proposed by Labourie in [25], together with its natural higher-dimensional generalisation.
We prove two injectivity theorems for the geodesic ray transform on two-dimensional, complete, simply connected Riemannian manifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with bounded non-positive curvature and the second with decaying non-positive curvature.
We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}\leq b<0$
We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.
Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
利用Chern-Gauss-Bonnet定理，在零化指标至少为n-3的Cartan-Hadamard流形M^n中，建立了凸超曲面的总Gauss-Kronecker曲率的尖锐不等式，从而将欧氏等周不等式推广到M^n，证明了这些空间的Cartan-Hadamard猜想。
We provide elementary proofs of Lemmas 7.1 and 7.4 appearing in "The Cartan-Hadamard conjecture and the Little Prince", by B. Kloeckner and G. Kuperberg. The Lemmas play an important role in the derivation of novel isoperimetric inequalities. The original proofs relied on Sage, a symbolic algebra package, to factor certain algebraic varieties into irreducible components.
If $(N^{m+p},h)$ is a Cartan-Hadamard manifold such that $Ric(h)\geq -G(r_{N}(x))$ where $G(0)\geq 1, G^{'}\geq 0$ and $G^{-1/2}\not\in L^{1}(+\infty)$ then every proper biharmonic isometric immersion $ϕ: M^{m}\rightarrow(N^{m+p},h)$ is a harmonic map.
Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition of potential function is isometric to the Euclidean space. Next we have proved a compactness theorem for gradient shrinking Ricci soliton satisfying some scalar curvature condition. Finally, we have showed that a gradient expanding Ricci soliton with linear volume growth and positive potential function is an Einstein manifold.
We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds.
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。
Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds
本文针对具有挤压负曲率的Cartan-Hadamard流形，建立凸超曲面总Gauss-Kronecker曲率的尖锐下界，结合Chern-Gauss-Bonnet定理等完成证明，还应用该结果得到5维中Cartan-Hadamard猜想的等周不等式。
Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds
该研究证明了Cartan-Hadamard流形中凸超曲面的总曲率下界，扩展了Borbély定理并给出了体积下界。
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.
Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds
本文在旋转对称 Cartan-Hadamard 流形上引入全局频率函数，并在实双曲空间中证明相关频率函数的单调性，进而得到对数凸性定理和双曲三球定理。
A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound
本文在 Ricci 曲率下界条件下证明了小体积情形的广义 Cartan-Hadamard 猜想，并给出任意体积情形向等式刚性归约的关键简化。
The uniqueness of minimal maps into Cartan-Hadamard manifolds via the squared singular values
该研究利用平方奇异值相关函数的凸性，证明了非正曲率黎曼流形上极小映射Dirichlet问题的唯一性定理，改进了已有文献中的相关结果。
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 introduce a family of functionals on submanifolds of Cartan-Hadamard manifolds that generalize the Colding-Minicozzi entropy of submanifolds of Euclidean space. We show that these functionals are monotone under mean curvature flow under natural conditions. As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon.
Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces
本文通过调和平均曲率流在Cartan-Hadamard 3空间中证明了锐利的Minkowski型不等式，改进了超曲面总平均曲率的已知估计，特别改进了Ghomi-Spruck的结果。
Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds
研究Cartan-Hadamard流形中总绝对曲率最小的闭曲面，证明其边界为平坦凸体，并推广Chern-Lashof定理，解决Gromov于1985年提出的问题。
It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension $ n\ge 3 $, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. As a byproduct of the Cartan-Hadamard conjecture, a longstanding problem in the mathematical literature settled only very recently in a breakthrough paper by Ghomi and Spruck, we can now assert that the optimal constant is also Euclidean, namely it coincides with the one achieved in the Euclidean space $ \mathbb{R}^n $ by the Aubin-Talenti functions. One may ask whether there exist at …
The generalized Cartan-Hadamard conjecture says that if $Ω$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le κ\le 0$, then the boundary of $Ω$ has the least possible boundary volume when $Ω$ is a round $n$-ball with constant curvature $K=κ$. The case $n=2$ and $κ=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $κ=0$, and a special case of the conjecture for $κ\textless{} 0$ and a version for $κ\textgreater{} 0$. Our argument uses a new interpretation, based on opt…
We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature.
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 give a general criterion for the Dirichlet problem at infinity (DPI) on a Cartan-Hadamard surface to be solvable, which we primarily use to give the best possible upper radial radial curvature bound for solvability of the DPI, but which is also flexible enough to accommodate flats. In particular, any (upper) radial curvature bound which implies transience also implies solvability of the DPI, which is perhaps surprising. Taking advantage of the structure provided by uniformization, we show that solvability of the DPI implies there is a natural continuous surjection of the Martin boundary ont…
We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive curvature. We also prove a modified version of Croke's isoperimetric inequality for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature. We discuss the relationship between these results, and…
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。
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…
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.
The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature
本文在截面曲率受控变化条件下证明了任意维度的Cartan-Hadamard猜想，通过控制商极小化子的凸包并比较度量畸变与双曲等周轮廓盈余，给出关键证明步骤。
Minkowski inequality in Cartan-Hadamard manifolds
本文利用调和平均曲率流证明 Cartan-Hadamard 三维流形中给定面积凸曲面的精确总平均曲率下界，并推出非正曲率空间中的 Bonnesen 型等周不等式，且将联系推广至任意维数。
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…
On the Dirichlet boundary value problem on Cartan-Hadamard manifolds
本文研究Cartan-Hadamard流形上半线性椭圆方程的Dirichlet问题，通过新的比较技术揭示曲率、拉普拉斯谱与渐近边界几何的相互作用。
We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds, and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.
Reilly-type inequalities for submanifolds in Cartan-Hadamard manifolds
本文研究了Cartan-Hadamard流形中子流形的特征值估计问题，提出了p-Laplacian和$L_T$算子第一个非零特征值的新上界，推广并改进了已有结果，并导出了加权流形与边值问题的Reilly型不等式。
The Cartan-Hadamard conjecture in dimension five
本文证明了五维完备单连通非正截面曲率流形上的欧几里得等周不等式，确立了Cartan-Hadamard猜想，方法基于边界点对积分与Jacobi场估计，并推广至三维。
Fourth order Schrödinger equation with mixed dispersion on certain Cartan-Hadamard manifolds
本文研究Cartan-Hadamard流形上混合色散四阶薛定谔方程，在双曲空间证明小初值全局解与散射，在旋转对称流形上获得加权Strichartz估计及爆破结果。
A synthetic Lorentzian Cartan-Hadamard theorem
本文提出并证明了一个合成洛伦兹Cartan-Hadamard定理，将局部凸度量空间的结论扩展到洛伦兹几何框架，并研究了时空测地线的存在性与唯一性。
Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds
本文证明Cartan-Hadamard流形中曲率在切平面消失的闭无穷小凸超曲面围成凸平坦区域，推广了经典刻画与刚性结果，并应用Gauss-Codazzi方程和Alexandrov几何技巧。
In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we show that (1) $\partial ^\infty \tilde M_1^n$ is homeomorphic to $\partial ^\infty \tilde M_2^n$, and (2) $\tilde M_1^n$ is homeomorphic to $\tilde M_2^n$. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension $\geq 6$ are topologically rigid. We include some straightforward consequences of topological rigidity (diagra…
We show how a Bochner type formula can be used to establish universal inequalities for the eigenvalues of the drifted Cheng-Yau operator on a bounded domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary condition. In the first theorem, the hyperbolic space case is treated in an independent way. For the more general setting, we first establish a Rauch comparison theorem for the Cheng-Yau operator and two estimates associated with the Bochner type formula for this operator. Next, we get some integral estimates of independent interest. As an application, we compute our univers…
We show, by modifying Borbély's example, that there are $3$-dimen\-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $\le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.
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 Cartan-Hadamard conjecture states that, on every $n$-dimensional Cartan-Hadamard manifold $ \mathbb{M}^n $, the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for $n \le 4$. It was also shown that its validity in dimension $n$ ensures that every $p$-Sobolev inequality ($ 1 < p < n $) holds on $ \mathbb{M}^n $ with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal fun…
The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds
本文通过常平均曲率超曲面的尖锐不等式及边界点对积分和Jacobi场估计，证明了$3\leq n\leq9$维Cartan-Hadamard流形中的欧几里得等周不等式，确立了相应维度的Cartan-Hadamard猜想。
Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds
本文利用两个初等泛函不等式，在 Cartan-Hadamard 流形上建立了高阶算子（含 clamped 板和 buckling 板问题）的尖锐谱间隙估计，解决了 Cheng-Yang 的渐近问题并回答了 Kristály 的提问，同时得到 Rellich 不等式。
Coupled nonlinear integrable systems are generated from usual zero curvature equation. The relevant Maurer-Cartan forms are constructed by combining suitably chosen matrices (nilpotent, Hadamard, idempotent and k-idempotent) and Lie algebraic elements via Kronecker product. In each case a closure type property among the matrices chosen is found to be playing a key role to produce both the coupling and nonlinearity present in the system of equations obtained. The method is highly flexible and can be used to construct general systems containing 'p' number of equations. It is also shown that thes…
We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If $p<m$, large data blow up at worst in infinite time, and under the stronger restriction $p\in(1,(1+m)/2]$ all data give rise to solutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The results are in several aspects …
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。
Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $Σ$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp. flat chain mod 2) such that $\partial S = Σ$. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $Σ$, to show that S satisfies the optimal Euclidean isoperimetric inequality: $ 6 \sqrtπ\, \mathbf{M}[S] \leq (\mathbf{M}[Σ])^{3/2} $. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from ab…
Unexpected Analytic Phenomena on Finsler Manifolds
本文研究平坦Finsler Cartan-Hadamard流形上的分析性质，发现Nash嵌入定理失效、Sobolev空间非线性、Hardy和不确定性不等式完全失效，而CKN不等式存在参数依赖的尖锐阈值，并归因于S-曲率下界。
We give a survey on the development of the study of the asymptotic Dirichlet problem for the minimal surface equation on Cartan-Hadamard manifolds. Part of this survey is based on the introductory part of the doctoral dissertation of the author. The paper is organised as follows. First we introduce Cartan-Hadamard manifolds and the concept of asymptotic Dirichlet problem, then discuss about the development of the results and describe the methods used in the proofs. In the end we mention some results about the nonsolvability of the asymptotic Dirichlet problem.
Rigidity and existence of Busemann profiles for the Allen--Cahn equation on Cartan--Hadamard manifolds
该研究在Cartan-Hadamard流形上探讨Allen–Cahn方程的Busemann剖面，证明了相关刚性原理，明确了双稳不平衡势下Busemann前沿存在的条件，并在特定曲面上验证了结果的尖锐性。
We use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard space, we formulate criterions for the action to have a global fixed point.
Global minimizers for fast diffusion versus nonlocal interactions on negatively curved manifolds
本文在Cartan-Hadamard流形上研究含快速扩散与非局部相互作用的自由能泛函，给出全局极小化子存在的充要条件，并推导新的Carlson-Levin型不等式。
Let G be a compact, connected Lie group, acting smoothly on a manifold M. Goresky-Kottwitz-MacPherson described a small Cartan model for the equivariant cohomology of M, quasi-isomorphic to the standard Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small Cartan model into the standard Cartan model, inducing an isomorphism in cohomology. The construction involves the solution of a remarkable inhomogeneous Maurer-Cartan equation. This solution has further applications to the theory of transgression in the Weil algebra, and to the C…
We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive several geometric inequalities for a convex hypersurface $Γ$ in a Cartan-Hadamard manifold $M$. In particular we show that the first mean curvature integral of a convex hypersurface $γ$ nested inside $Γ$ cannot exceed that of $Γ$, which leads to a sharp lower bound in dimension $3$ for the total first mean curvature of $Γ$ in terms of the volume it bounds in $M$. This monotonicity property is extended to all mean curvature integrals when $γ$ is paralle…
Gromov-Hausdorff Stability and Rigidity of manifolds via Heisenberg-Pauli-Weyl Uncertainty Principle
本文利用海森堡-泡利-魏尔不确定性原理，研究黎曼流形的格罗莫夫-豪斯多夫稳定性与刚性，确立了不同曲率流形的收敛性，修正了现有HPW不等式并给出定量刚性估计。
We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is …
We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -κ^2$ for some $κ\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounded from below by $\frac{(n-1)^4}{16}κ^4$ whenever the $κ$-Cartan-Hadamard conjecture holds on $(M,g)$, e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped pl…
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