Lipschitz-volume rigidity of Lipschitz manifolds among integral currents
Roger Züst
本文证明了在充分条件下，从度量积分流形到无穷小欧几里得Lipschitz流形的保体积1-Lipschitz映射为等距映射。
我们给出充分条件，使得从度量积分流形到无穷小欧几里得Lipschitz流形的保体积1-Lipschitz映射是等距映射。
We give sufficient conditions such that a volume preserving 1-Lipschitz map from a metric integral current onto an infinitesimally Euclidean Lipschitz manifold is an isometry.
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Let $X$ and $Y$ be length metric spaces. Let $\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map $f\colon X\to Y$ and $0<\mathcal H^n(X)=\mathcal H^n(f(X))<\infty$, then $f$ preserves the length of path. This property holds for smooth manifolds but doesn't hold for all singular spaces. We survey the Lipschitz-Volume Rigidity Theorems on singular spaces with lower curvature bounds and discuss some related open problems.
The Lipschitz-volume rigidity problem for metric manifolds
本文研究度量流形与 Riemann 流形间非零度 1-Lipschitz 映射的体积刚性，借助度理论和积分流刚性的新进展证明了相应结果。
We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with respect to $X$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.
We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.
We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-Émery Ricci curvature.
Rigidity of mass-preserving $1$-Lipschitz maps from integral current spaces into $\mathbb{R}^n$
本文证明了从$n$维积分流空间到欧氏球的保质量且边界单射的$1$-Lipschitz映射必为等距映射，并由此导出内蕴平坦距离下的稳定性结果，蕴含图流形正质量定理的稳定性。
For smooth surfaces properly immersed in the unit ball of $\RR^n$ with density close to one and small Willmore energy, the optimal a priori estimate(bi-Lipschitz and $W^{2,2}$ parametrization)is provided. We also discuss the quantitative rigidity for $L^2$-almost CMC surfaces.
Lipschitz rigidity for scalar curvature on singular manifolds in odd dimensions
本文研究标量曲率在奇数维奇异流形上的Lipschitz刚性，结合谱流论证和抽象锥算子分析，证明了奇数维Riemannian spin流形的刚性结果。
Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces
本文证明闭度量曲面到同面积闭黎曼曲面的1-Lipschitz满射必为等距映射，建立度量曲面上连续Sobolev函数的余面积不等式（推广Esmayli-Ikonen-Rajala结果），并研究非光滑目标空间下映射的正则性。
Lipschitz rigidity for scalar curvature
本文研究标量曲率的Lipschitz刚性问题，通过低正则性度量下扭曲Dirac算子的谱性质，结合拟正则映射理论与指标定理，推广了已有结果并正面解答了Gromov的相关问题。
Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces
本文证明了凸体在积分流空间中是其边界度量的唯一极小填充，并具有Lipschitz-体积刚性性质，同时回答了关于Plateau问题极小化序列内蕴平坦收敛的Perales问题。
Abstract cone operators and Lipschitz rigidity for scalar curvature on singular manifolds
本文利用扭曲Dirac算子的指标理论，证明奇异流形上数量曲率的Llarull型比较结果，并将Lipschitz刚性从偶数维推广到奇数维。
Motivated by the Lipschitz rigidity problem in scalar curvature geometry, we prove that if a closed smooth spin manifold admits a distance decreasing continuous map of non-zero degree to a sphere, then either the scalar curvature is strictly less than the sphere somewhere or the map is a distance isometry. Moreover, the property also holds for continuous metrics with scalar curvature lower bound in some weak sense. This extends a result in the recent work of Cecchini-Hanke-Schick and answers a question of Gromov. The method is based on studying the harmonic map heat flow coupled with the Ricci…
Sharp bounds and rigidity for volumes of boundaries of Alexandrov spaces
该研究针对具有给定曲率下界、维度和半径的Alexandrov空间，推导了其边界体积的精确界，完成刚性情形分类与几乎刚性分析，成果对带边光滑流形也具创新性。
Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $Z/\sim$, where $z, z'\in\partial Z$ and $z\sim z'$ if only if $f(z)=f(z')$. We will partially verify this conjecture, and give a classification for compact Alexandrov $n$-spaces with relatively maximal volume. We will also give an elementary proof for a pointed version of Bishop-…
Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthogonal, parallel Jacobi field. As applications we obtain (1) constraints on the behavior of quasi-isometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splitting…
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 …
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.
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.
Conjugate radius, volume comparison and rigidity
本文研究 Kähler 流形在不同曲率条件下的几何性质，建立共轭半径估计、体积比较与刚性定理。
Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.
New Volume Comparison Results and Volume Growth Rigidity of Gradient Ricci Almost Solitons
本文针对以ρ(x)为Bakry-Émery Ricci曲率下界的完备流形建立新体积比较定理，将Cao与Zhou的结果推广到收缩梯度Ricci几乎孤子，得到梯度Ricci几乎孤子的体积刚性及极大体积增长刚性结果。
We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak--Wenger, which satisfies a related maximality condition.
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.
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.
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].
This paper generalizes D. Burago and S. Ivanov's work on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of complex, quaternionic and Cayley hyperbolic metrics.
Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds
本文针对高维黎曼流形中的体积极小自由边界超曲面，建立了基于数量曲率和边界第二基本形式的体积估计，并在等式情形下证明了流形的刚性。
In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature.
Sobolev-to-Lipschitz property of geodesically complete spaces with curvature bounded from above
证明每个曲率上有界且测地完备的长度空间具有指数无穷的Sobolev-to-Lipschitz性质，即每个W^{1,∞}空间中的Sobolev映射有Lipschitz代表，且Lipschitz常数等于映射的无穷能量。
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 $\geq 3$.
The main subject of this expository paper is a connection between Gromov's filling volumes and a boundary rigidity problem of determining a Riemannian metric in a compact domain by its boundary distance function. A fruitful approach is to represent Riemannian metrics by minimal surfaces in a Banach space and to prove rigidity by studying the equality case in a filling volume inequality. I discuss recent results obtained with this approach and related problems in Finsler geometry.
This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.
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.
The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Secondly, we apply this almost rigidity theorem to study noncompact RCD(0, N) spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence of harmonic functions with polynomial growth on such RCD(0,N) sp…
In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of Kewei Zhang and Yuchen Liu on holomorphic rigidity of such Kahler manifolds with the structure theorem of Tian-Wang for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume
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.
Scalar-mean rigidity theorem for compact manifolds with boundary
本文通过发展平均曲率的降维论证，证明了维度小于五的带边界紧Riemann流形的标量-平均刚性定理，并得出精确球面半径刚性、最佳NNSC填入及Lipschitz Listing型刚性等推论。
Let $M$ be a compact Riemannian manifold with boundary. We show that $M$ is Gromov-Hausdorff close to a convex Euclidean region $D$ of the same dimension if the boundary distance function of $M$ is $C^1$-close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance function of $M$ is $C^0$-close to that of $D$, the volumes of $M$ and $D$ are almost equal, and volumes of metric balls in $M$ have a certain lower bound in terms of radius.
We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we obtain a volume growth rigidity theorem. We also show a splitting theorem of Cheeger-Gromoll type under the assumption of the existence of a single ray.
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manifolds with nonnegative and quadratically decaying lower Ricci curvature bounds have minimal volume growth. Compact regions in such manifolds are shown to be asymptotically close to warped products in the Gromov-Hausdorff topology. Manifolds with nonnegative Ricci curvature and lin…
Geodesic balls in a simply connected space forms $\mathbb{S}^n$, $\mathbb{R}^{n}$ or $\mathbb{H}^{n}$ are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that the boundary of the manifold is an Einstein hypersurface. In the same spirit we also extend a rigidity theorem due to Boucher et al. \cite{Bou} and Shen \cite{Shen} to $n$-dimensional static metrics with positive constant scalar curvature, which provides another proof of a parti…
In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $Σ$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of $Σ$. Furthermore, if $Σ$ saturates the respective upper bound and $M$ has nonnegative Ricci curvature, then $Σ$ is isometric to $\mathbb{S}^4$ up to scaling and $M$ splits in a neighborhood of $Σ$. Also, we obtain a rigidity result for the Riemannian…
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 prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.
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…
In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.
Length distortion of volume-preserving Lipschitz mappings
本文证明度量曲面间保面积Lipschitz映射几乎不畸变曲线长度，给出两种证明方法，并推广到高维及非光滑情形。
This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $M$ is bounded for $H\ne 1$. In [CRX], we verified the conjecture for the case that $M$ or its Riemannian universal covering space $\tilde M$ is not collapsed for $H=1$ o…
Volume comparison by timelike Lipschitz maps
本研究改进了洛伦兹预长度空间上的类时Hausdorff测度V_N得到W_N，通过保因果性与类时Lipschitz映射建立二者的体积比较不等式，证明其在光滑时空等空间中一致，并构造相关映射实例。
Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity
研究三维保体积阿诺索夫流，证明若两流拓扑共轭则共轭光滑，除非为常屋顶悬挂流；并推出曲面负曲率加权标记长度谱刚性等新结果。
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。
We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and distributional Laplacian comparison theorem on limit spaces of Riemannian manifolds.
We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two manifolds differ by less than a uniform constant after an appropriate normalization of the metrics. The results are qualitatively sharp in the sense that the dependencies are necessary. We give a number of corollaries.
A Finsler metric is geodesically reversible if geodesics remain geodesics after a change of orientation. Asymmetric norms on vector spaces and Funk metrics in the interior of convex bodies are examples of geodesically reversible metrics that are not necessarily sums of reversible metrics and closed 1-forms. However, there seem to be few such examples in closed manifolds. In this paper the theory of volumes and areas on Finsler spaces is applied to establish a number of rigidity theorems which partially explain this paucity of examples. These rigidity results settle some hitherto unsolved cases…
Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound
本文研究低Ricci曲率下界下的体积空间形式刚性问题，证明近似满足最大体积条件的紧致流形与常曲率空间形式微分同胚且在Gromov-Hausdorff拓扑下接近，改进并推广了微分同胚球面定理。
Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature
该研究针对正里奇曲率流形，证明了其尖锐梯度估计的几乎刚性，建立梯度亏格与体积亏格的定量控制关系，还得到爱因斯坦流形的间隙定理及闭4维爱因斯坦流形的刚性定理。
In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature $-b^2\le K\le -1$ and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely generated fundamental group with empty dissipative set to negative variable curvature 3-manifolds. We also generalize the rigidity of Hamenstädt or more recently Besson-Courtois-Gallot, to 3-manifolds with infinite volume and geometrically infinite fundamental group.
In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the $L$-stability operator and discuss the $L$-stability of some special self-expanders. Besides, we prove that the surfaces $Γ\times\…
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