Christopher Connell, Benson Farb
We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as well.
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A higher rank rigidity theorem for convex real projective manifolds
本文研究凸实射影流形的刚性问题，将 Ballmann 与 Burns-Spatzier 的高阶刚性定理推广到该情形，证明了相应类似定理。
We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions $\geq 3$, generalizing the work of arXiv:1708.03983 to the finite volume setting. The rigidity theorem uses the techniques of Besson, Courtois, and Gallot's entropy rigidity theorem. It implies uniform lower bounds on the volume of any finite volume strictly convex projective manifold in dimensions $\geq 3$.
For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson--Courtois--Gallot's entropy rigidity result to Hilbert geometries.
We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with $π_1$-injective cusps. We prove smooth (self) rigidity for this class of manifolds in the case where the gluing preserves the cusps' homogeneous structure. We compute the group of self homotopy equivalences of such a manifold and show that it can contain a normal free abelian subgroup and thus, can be infinite. Elements o…
We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given $N, D$, there exists $ε(N, D)>0$, such that for $ε<ε(N, D)$, if $X$ is an $N$-dimensional Alexandrov space with curvature $\geq -1$, $\operatorname{diam}(X)\leq D, h(X)\geq N-1-ε$, then $X$ is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity of \cite{CRX} to Alexandrov spaces. And we will also give a quantitative maximal volume entropy rigidity for $\op{RCD}^*$-spaces in the non-collapsing case.
We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.
In this note we give a short proof to the rigidity of volume entropy. The result says that for a closed manifold with Ricci curvature bounded from below, if the universal cover has maximal volume entropy, then it is the space form. This theorem was first proved by F. Ledrappier and X. Wang in [1].
We show that an $n$-dimensional surface whose entropy is close to that of an $n$-dimensional plane is close in Hausdorff distance to some $n$-dimensional plane at every scale. Moreover we show that self-expanders of low entropy converge in the Hausdorff sense to their asymptotic cones.
Let $M$ be a Riemannian manifold with dimension greater or equal to $3$ which admits a complete, finite-volume Riemannian metric $g_0$ locally isometric to a rank-1 symmetric space of non-compact type. The volume entropy rigidity theorem asserts that $g_0$ minimizes a normalized volume growth entropy among all complete, finite-volume, Riemannian metric on $M$. We will repair a gap in the proof when $g_0$ is locally isometric to the Cayley hyperbolic space.
We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities
We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.
We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds.
A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.
In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.
We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds.
Compactness and rigidity of self-shrinking surfaces
本文利用测度论技术与刚性结果，证明了低熵自收缩曲面的紧致性定理，进而确立熵极小化子的存在性并改进刚性结论。
In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincaré exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang.
We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded nonpositive flag curvature with rank at least $2$ whose universal cover is irreducible, is a locally symmetric space or a locally Minkowski space.
Let f:(Y,g)->(X,g_0) be a non zero degree continuous map between compact Kähler manifolds of dimension greater or equal to 2, where g_0 has constant negative holomorphic sectional curvature. Adapting the Besson-Courtois-Gallot barycentre map techniques to the Kähler setting, we prove a gap theorem in terms of the degree of f and the diastatic entropies of (Y, g) and (X,g_0), which extends the rigidity result proved by the author in [13].
We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.
Entropy collapse versus entropy rigidity for Reeb and Finsler flows
该研究对比切触流形上Reeb流的拓扑熵可任意小的熵坍缩现象，与许多闭流形上归一化Finsler测地流拓扑熵有一致正下界的熵刚性性质，揭示两类流的熵行为差异。
If a group $Γ$ acts geometrically on a CAT(0) space $X$ without 3-flats, then either $X$ contains a $Γ$-periodic geodesic which does not bound a flat half-plane, or else $X$ is a rank 2 Riemannian symmetric space, a 2-dimensional Euclidean building or non-trivially splits as a metric product. Consequently all such groups satisfy a strong form of the Tits Alternative.
Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic Kähler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary, when $X=Y$, we show that the minimal diastatic entropy is achieved if and only if $g$ is holo…
The volume entropy of a compact metric measure space is known to be the exponential growth rate of the measure lifted to its universal cover at infinity. For a compact Riemannian $n$-manifold with a negative lower Ricci curvature bound and a upper diameter bound, it was known that it admits an almost maximal volume entropy if and only if it is diffeomorphic and Gromov-Hausdorff close to a hyperbolic space form. We prove the quantitative rigidity of almost maximal volume entropy for $\operatorname{RCD}$-spaces with a negative lower Ricci curvature bound and Riemannian manifolds with a negative …
Curvature-dimension condition, rigidity theorems and entropy differential inequalities on Riemannian manifolds
本文通过信息论方法研究黎曼流形上的曲率-维度条件、刚性定理和熵微分不等式，提供了新的等价特征化和流形特征化。
Entropy Rigidity for Maximal Representations
本文研究了最大表示下的熵刚性问题，通过引入可测的格罗莫夫积和相关测度，证明了最大表示在熵方面的刚性性质。
In this note we will show the almost maximal volume entropy rigidity for manifolds with lower integral Ricci curvature bound in the non-collapsing case: Given $n, d, p>\frac{n}{2}$, there exist $δ(n, d, p), ε(n, d, p)>0$, such that for $δ<δ(n, d, p)$, $ε<ε(n, d, p)$, if a compact $n$-manifold $M$ satisfies that the integral Ricci curvature has lower bound $\bar k(-1, p)\leq δ$, the diameter $diam(M)\leq d$ and volume entropy $h(M)\geq n-1-ε$, then the universal cover of $M$ is Gromov-Hausdorff close to a hyperbolic space form $\Bbb H^k$, $k\leq n$; If in addition the volume of $M$, $vol(M)\geq…
By means of a space-time Wasserstein control, we show the monotonicity of the W-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the W-entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian m…
We say that a Riemannian manifold M has rank at least k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns-Spatzier, later generalized by Eberlein-Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank at least 2 (the higher rank assumption), whose isometry group satisfies the condition that the recurrent vectors are dense in the unit tangent bundle SM (the duality condition), is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. W…
We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME rigidity of higher rank lattices: any countable group which is ME to a lattice in a simple Lie group G of higher rank, is commensurable to a lattice in G.
Let $Γ$ be a non-uniform lattice in $PU(p,1)$ without torsion and with $p\geq2 $. We introduce the notion of volume for a representation $ρ:Γ\rightarrow PU(m,1)$ where $m \geq p$. We use this notion to generalize the Mostow--Prasad rigidity theorem. More precisely, we show that given a sequence of representations $ρ_n:Γ\rightarrow PU(m,1)$ such that $\lim_{n \to \infty} \text{Vol}(ρ_n) =\text{Vol}(M)$, then there must exist a sequence of elements $g_n \in PU(m,1)$ such that the representations $g_n \circ ρ_n \circ g_n^{-1}$ converge to a reducible representation $ρ_\infty$ which preserves a to…
This paper presents a rank rigidity result for negatively curved spaces. Let $M$ be a compact manifold with negative sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-Λ^2 < K < -λ^2$ where $λ/Λ> .93$. When $a$ is extremal, i.e. $-a^2$ is the curvature minimum or maximum for the manifold, this result is analogous to rank rigidity resu…
For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in $\mathsf{RCD}^{\ast}$ spaces. As an application we obtain an almost rigidity result which partially recovers a result by Cheng-Rong-Xu for Riemannian manifolds.
Rigidity of compact rank one symmetric spaces
本文证明了截面曲率在0到1之间的黎曼度量在凸真子集外与原度量相等时必等距于原度量，补充了紧秩一对称空间的刚性结果。
We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper "non-stationary" generalization of the classical the…
Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank, then M is isometric to a compact, rank one symmetric space (CROSS) i.e., isometric to a sphere, complex projective space, quaternionic projective space or to the Cayley plane. The notion of spherical rank is analogous to the notions of Euclidean rank and hy…
Entropy rigidity for cusped Hitchin representations
本文针对几何有限Fuchsian群的带尖点Hitchin表示建立熵刚性定理，引入(1,1,2)-超横截群类与射影可见群横截表示理论，推广了多个相关已有结论。
Volume entropy and rigidity for RCD-spaces
本文发展BCG重心技术至RCD空间，证明熵-体积不等式及刚性结果，并推广Lipschitz体积刚性至RCD$(K,N)$空间，应用于4-orbifolds上的爱因斯坦度量。
In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $λ_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrappier-Wang \cite{LeW2010volent} proved that the universal cover $\tilde{M}$ is isometric to the hyperbolic space $\mathbb{H}^n$. We will prove analogue theorems for Alexandrov spaces.
This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-Λ^2 < K < -λ^2$ where $λ/Λ> >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity t…
We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuchsian.
Given a convex representation $ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$ of a convex co-compact group $Γ$ of $\mathbb{H}^k$ we find upper bounds for the quantity $αh_ρ,$ where $h_ρ$ is the entropy of $ρ$ and $α$ is the Hölder exponent of the equivariant map $\partialΓ\to\mathbb{P}(\mathbb{R}^d).$ We also give rigidity statements when the upper bound is attained. We then study Hitchin representations and prove that if $ρ:π_1Σ\to\textrm{PSL}(d,\mathbb{R})$ is in the Hitchin component then $αh_ρ\leq 2/(d-1)$ (where $α$ is the Hölder exponent of the map $ζ:\partial\mathbb{H}^2\to\mathscr{F}$) with equalit…
In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on CAT$(-1)$-spaces that can be seen as a metric analog to the "entropy rigidity" theorems of U. Hamenstädt and M. Bourdon. Building on the ideas developed in \cite{BK02}, we establish a rigidity theorem for certain expanding quasi-Möbius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding …
De Rham $L^p$-Cohomology For Higher Rank Spaces And Groups: Critical Exponents And Rigidity
研究高秩李群和相关流形的L^p上同调临界指数及刚性特性，通过谱序列论证揭示非正曲率可解李群的连续准等距类。
Topological rigidity of small RCD(K,N) spaces with maximal rank
研究显示，小RCD(K,N)空间在最大秩条件下具有拓扑刚性，与infranilmanifold同胚，扩展了Kapovitch-Wilking的结果至非光滑环境。
Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $Γ<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $Γ$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $Γ$ by toral automorphisms is locally rigid. More generally, given a manifold $M$ on which $Γ$ acts isometrically and a torus $\Ta^n$ on which it acts by automorphisms, we show that the diagonal action on $\Ta^n{\times}M$ is locally rigid. This paper is the culmination of a series of papers and …
Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity
本文研究高维无尖点 Margulis 时空模空间，构造压力形式，借助带标记 Margulis 不变量谱的无穷小确定性证明其在恒定熵截面上诱导黎曼度量，并确立固定线性部分时这些截面的凸性。
Rigidity and gap theorems for Ricci shrinkers
本文证明了Ricci收缩体的局部Ricci曲率和ν-熵间隙定理，扩展了Munteanu-Wang和作者与Ma之前的成果，并提供了一种局部去除Ricci流I型奇点的标准。
We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian manifolds which are almost-isometric have the same volume growth entropy. We establish various rigidity results as applications.
Ballmann's Rank Rigidity Conjecture predicts that a CAT(0) space of higher rank with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a direct product. We confirm this conjecture in rank 2. For CAT(0) spaces of higher rank $n\geq 2$ we prove rigidity if the space contains a periodic $n$-flat and every complete geodesic lies in some $n$-flat. This does not require a geometric group action.
This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let $X$ be a CAT(0) space with a geometric group action. Suppose that every geodesic in $X$ lies in an $n$-flat, $n\geq 2$. If $X$ contains a periodic $n$-flat which does not bound a flat $(n+1)$-half-space, then $X$ is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.
We prove the holomorphic rigidity conjecture of Teichmüller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichmüller space as a complex manifold. The method of proof is through harmonic maps. We prove that the singular set of a harmonic map from a smooth $n$-dimensional Riemannian domain to the Weil-Petersson completion $\overline{\mathcal T}$ of Teichmüller space has Hausdorff dimension at most $n-2$, and moreover, $u$ has certain decay near the singular set. Combining this with the earlier work of Schumacher, Siu and Jost-Yau, we prov…
The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.
In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the rigidity models of the Shannon entropy power are Einstein or quasi Einstein manifolds with Hessian solitons. Moreover, we prove the convexity of the Shannon entropy power for the conjugate heat equation introduced by G. Perelman on Ricci flow and that the corresponding rigidity mod…
Rigidity of ideal symmetric sets
本文利用无穷远处对径集，在哈达玛空间中刻画了高秩模型几何（黎曼对称空间、欧几里得建筑和乘积空间）的刚性特征。
Schwarz lemmas and rigidity of \(\bar\partial_b\)-harmonic maps
本文证明从完备伪厄米流形到负曲率Kähler流形的\\(\bar\partial_b\\)-调和映射的Schwarz引理，并利用Bochner恒等式和Siu型散度恒等式，在Sasakian源和复双曲目标上分类达到水平界的映射，导出叶状结构、多重调和性和秩刚性。
In this paper, we study degenerate CR embeddings $f$ of a strictly pseudoconvex hypersurface $M\subset \bC^{n+1}$ into a sphere $\bS$ in a higher dimensional complex space $\bC^{N+1}$. The degeneracy of the mapping $f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings $f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained a complex plane section …
I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.
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