Vitali Kapovitch, Christian Ketterer, Karl-Theodor Sturm
In this paper we prove that in the class of metric measure spaces with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD(K,N)$ with $K\in \mathbb{R}$ and $N\in [1,\infty)$ is preserved under doubling and gluing constructions.
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In this paper we discuss the sufficient and necessary conditions for multiple Alexandrov spaces being glued to an Alexandrov space. We propose a Gluing Conjecture, which says that the finite gluing of Alexandrov spaces is an Alexandrov space, if and only if the gluing is by path isometry along the boundaries and the tangent cones are glued to Alexandrov spaces. This generalizes Petrunin's Gluing Theorem. Under the assumptions of the Gluing Conjecture, we classify the $2$-point gluing over $(n-1,ε)$-regular points as local separable gluing and the gluing near un-glued $(n-1,ε)$-regular points a…
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $\mathbb{R}P^2$. We then classify closed three-dimensional $\mathsf{CD}^*(0,3)$-Alexandrov spaces.
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.
Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.
Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we generalize a result by Colding that an n dimentional manifold with Ricci curvature greater or equal to n minus 1 and volume close to that of the unit n sphere is close (in the Gromov-Hausdorff distance) to the sphere, from the case of Riemannian manifolds to the case of Alexandrov spaces, with an additional assumption, roughly speaking, that the rough volume of the set of short cut points is small, following the basic idea in the Riemannian case with necessary modifications because of the on…
Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature $\geq κ$ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the $\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane, which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.
In this paper, we show existence and uniqueness of Ricci flow whose initial condition is a compact Alexandrov surface with curvature bounded from below. This requires a weakening of the notion of initial condition which is able to deal with a priori non-Riemannian metric spaces. As a by-product, we obtain that the Ricci flow of a surface depends smoothly on Gromov-Hausdorff perturbations of the initial condition.
In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.
We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem.
Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery) Ricci curvature.
We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below. For the proof we combine techniques developed in these papers. Because the Finsler product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again …
Here I show compatibility of two definition of generalized curvature bounds --- the lower bound for sectional curvature in the sense of Alexandrov and lower bound for Ricci curvature in the sense of Lott--Villani--Sturm.
On the uniqueness of Ricci flows from certain Alexandrov spaces
本研究证明从奇异Reifenberg空间出发、满足尺度不变曲率界等条件的Ricci流唯一性，并据此在极限空间上规范定义光滑结构，推广了Perelman稳定性定理。
Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positive Ricci curvature.
In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature.
An invitation to Alexandrov geometry: CAT(0) spaces
本文通过最少的准备，展示 Alexandrov 几何的美丽与强大，涵盖 CAT(0) 空间的多个应用和定理。
Geometric inequalities and the Alexandrov-Bakelman-Pucci technique
本文基于Alexandrov-Bakelman-Pucci技术，统一讨论了几何不等式的证明框架，涵盖欧几里得空间的等周不等式、子流形的均曲率不等式及Sobolev不等式等核心方法与贡献。
In this paper we construct Ricci-positive metrics on the connected sum of products of arbitrarily many spheres provided the dimensions of all but one sphere in each summand are at least 3. There are two new technical theorems required to extend previous results on sums of products of two spheres. The first theorem is a gluing construction for Ricci-positive manifolds with corners generalizing a gluing construction of Perelman for Ricci-positive manifolds with boundaries. Our construction gives a sufficient condition for gluing together two Ricci-positive manifolds with corners along isometric …
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.
A generalization of the Perelman gluing theorem and applications
本文将Perelman正Ricci曲率粘合定理推广到从正标量曲率到正截面曲率的正中间曲率条件，并应用于证明观察者模空间具有非平凡高阶同伦群及给出带全测地边界的正中间Ricci曲率度量存在性充分条件。
We derive several vanishing theorems for genera under almost nonnegative Ricci curvature and infinite fundamental group, which includes Todd genus, $\widehat{A}$-genus, elliptic genera and Witten genus. A vanishing theorem of Euler characteristic number for almost nonnegatively curved Alexandrov spaces is also proved.
We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method of Cauchy data matching from \cite{MP}, cf. also \cite{MPP}. In the second part of this paper, we develop the consequences of this result and (at least partially) characterize the image of the map which associates to each complete, Alexandrov-embedded CMC surface with finite topology its associated conformal structure, which is a compact Riemann surface wit…
Glued spaces and lower curvature bounds
本文综述光滑与非光滑语境下受下曲率界约束的粘合构造定理并提出猜想，聚焦合成下Ricci曲率界，涉及多种空间类型，为相关研究提供理论梳理与猜想方向。
During the years 1940-1970, Alexandrov and the "Leningrad School" have investigated the geometry of singular surfaces in depth. The theory developed by this school is about topological surfaces with an intrinsic metric for which we can define a notion of curvature, which is a Radon measure. This class of surfaces has good convergence properties and is remarkably stable with respect to various geometrical constructions (gluing etc.). It includes polyhedral surfaces as well as Riemannian surfaces of class $C^2$, and both of these classes are dense families of Alexandrov's surfaces. Any singular …
We prove generalized lower Ricci bounds for Euclidean and spherical cones over complete Riemannian manifolds. These cones are regarded as complete metric measure spaces. In general, they will be neither manifolds nor Alexandrov spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0; n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n; n+1). More generally, for each N > 1 we prove that the condition CD(N-1;N) for a wei…
In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if…
We glue two manifolds which have curvature operators at least k (in the sense of eigenvalues) along their common boundary. We show that if the sum of the second fundamental forms of the boundary is positive semidefinite, then the curvature operator of the resulting manifold is at least k up to an arbitrarily small error term. Similar results hold for Ricci, scalar, bi, isotropic and flag curvature, respectively.
We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estimate for p larger than 1. It also has several applications in geometry, providing the missing step in order to pose the Ricci flow with rough initial data in the noncompact case, for example starting with a general noncompact Alexandrov surface, and giving the sharp asymptotics for the contracting cusp Ricci flow, as we show elsewhere.
We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces.
The metric measure boundary of spaces with Ricci curvature bounded below
本文证明无边界 RCD(K,N) 空间的度量测度边界为零，解决相关猜想，并与既有结果共同回答无边界 Alexandrov 空间中无限测地线的存在性问题。
Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds
本文针对具有下Ricci曲率界的非光滑RCD空间，通过原创变分估计与渐近质量分解方法，证明了尖锐等周比较定理与等周轮廓的维数凹性，推广了紧流形及欧氏凸体上的经典结果。
We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+εI \in C at the initial time, then it satisfies R +KεI \in C on some time interval depending only on the scalar curvature control. This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and R + I \in C along the sequence of initial conditi…
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-linear operators. These inequalities seem not available in the literature; And our proof, solely based on the ABP estimate, does not involve any Sobolev inequalities nor gradient estimate. We also propose a question regarding the characterization of Ricci lower bound by the Harnack …
The observer moduli space of Riemannian metrics is the quotient of the space $\mathcal{R}(M)$ of all Riemannian metrics on a manifold $M$ by the group of diffeomorphisms $\mathrm{Diff}_{x_0}(M)$ which fix both a basepoint $x_0$ and the tangent space at $x_0$. The group $\mathrm{Diff}_{x_0}(M)$ acts freely on $\mathcal{R}(M)$ providing $M$ is connected. This offers certain advantages over the classic moduli space, which is the quotient by the full diffeomorphism group. Results due to Botvinnik, Hanke, Schick and Walsh, and to Hanke, Schick and Steimle have demonstrated that the higher homotopy …
Ricci flow from singular spaces with bounded curvature
本文证明了以亚历山德罗夫意义下有界曲率的紧长度空间为初值的里奇流存在，并收敛到与原空间等距的 $C^{1,α}$ 黎曼流形，且解在等距意义下唯一。
Kähler-Ricci flows coming out of metric spaces
本文研究紧Kähler流形上闭正(1,1)-流诱导可作为Kähler-Ricci流零时刻Gromov-Hausdorff极限的度量结构的充分条件，并在1维情形拓展已有工作证明了对应Alexandrov曲面可作为Kähler-Ricci流初始数据。
Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature
本文利用Alexandrov-Bakelman-Pucci方法，将Michael-Simon型Sobolev不等式推广到具有非负中间Ricci曲率的Riemannian流形中的子流形上，并由此导出极小子流形的等周不等式。
In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow to prove a new class of geometric inequalities for horospherically convex hypersurfaces in hyperbolic space. Using these new Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in …
Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows
本文提出标量曲率（及其积分）的合成概念，通过高斯积分的初始斜率定义，并计算了Lipschitz粘合光滑黎曼流形和锥体的积分标量曲率。在二维情况下，前者与高斯-博内公式一致，后者则不同。时间依赖性的扩展可将里奇流表征为具有最小积分曲率功能的超里奇流。
Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of $\mathsf{RCD}^{…
A Kümmer construction for Chern-Ricci flat balanced manifolds
本文针对带有孤立奇点和Chern-Ricci平坦平衡度量的非Kähler Calabi-Yau轨形，通过粘合构造证明其所有crepant分解均存在Chern-Ricci平坦平衡度量，并探讨了在寻找Hull-Strominger系统解及奇异三维簇中的应用。
The theory of complete surfaces of (nonzero) constant mean curvature in $\RR^3$ has progressed markedly in the last decade. This paper surveys a number of these developments in the setting of Alexandrov embedded surfaces; the focus is on gluing constructions and moduli space theory, and the analytic techniques on which these results depend. The last section contains some new results about smoothing the moduli space and about CMC surfaces in asymptotically Euclidean manifolds.
Chern-Ricci flat balanced metrics on small resolutions of Calabi-Yau threefolds
本文通过粘合构造证明，射影节点 Kähler Calabi-Yau 三流形的所有小分解均容许 Chern-Ricci 平衡度量，从而解决了 Hull-Strominger 系统的伸缩子方程。
In this essay, we discuss the notion of optimal transport on geodesic measure spaces and the associated (2-)Wasserstein distance. We then examine displacement convexity of the entropy functional on the space of probability measures. In particular, we give a detailed proof that the Lott-Villani-Sturm notion of generalized Ricci bounds agree with the classical notion on smooth manifolds. We also give the proof that generalized Ricci bounds are preserved under Gromov-Hausdorff convergence. In particular, we examine in detail the space of probability measures over the interval, $P(X)$ equipped wit…
This paper completes a fundamental construction in Alexandrov geometry. Previously we gave a new construction of metric spaces with curvature bounds either above or below, namely warped products with intrinsic metric space base and fiber, and with possibly vanishing warping functions -- thereby extending the classical cone and suspension constructions from interval base to arbitrary base, and furthermore encompassing gluing constructions. This paper proves the converse, namely, all conditions of the theorems are necessary. Note that in the cone construction, both the construction and its conve…
Jacobian estimates and geometric inequalities under intermediate Ricci curvature
本文证明了一个适用于任意余维子流形的 Jacobian 估计，无需曲率符号假设，并由此在中间 Ricci 曲率条件下推导出定量 Fenchel--Willmore、Sobolev 和等周不等式，所需条件仅与子流形维数相关。
We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the $p$-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty)$. This extends to the non-smooth setting a classical result of Bér…
We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperimetric profile of the model space. Moreover, if equal…
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on $RCD^{*}(K,N)$ metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian mani…
Smoothing polyhedral spaces via Ricci flow
本文证明了Petrunin光滑化猜想：任意紧致无边界非负Alexandrov曲率的多面体空间是光滑黎曼轨形空间的Gromov-Hausdorff极限，通过新Ricci流短时间存在理论实现，并给出积分曲率缺陷的刚性应用。
The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K,N) metric measure spaces): - we develop an intrinsic theory of Laplacian bounds in viscosity sense and in a pointwise, heat flow related, sense, showing their equivalence also with Laplacian bounds in distributional sense; - relying on these tools, we establish a PDE principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the $p$-Hopf-Lax semigroup, for general exponents $p\in[1,\infty)$. This principle adm…
Boundary flexibility for curvature conditions
本文提出一个普遍边界粘合原理，统一了Perelman、Gromov-Lawson和Bär-Hanke的经典粘合定理，并借助代数曲率锥和局部灵活性引理，证明了一个边界变形原理，用于比较不同边界条件下的度量空间。
Given a closed manifold $M$ and a vector bundle $ξ$ of rank $n$ over $M$, by gluing two copies of the disc bundle of $ξ$, we can obtain a closed manifold $D(ξ, M)$, the so-called double manifold. In this paper, we firstly prove that each sphere bundle $S_r(ξ)$ of radius $r>0$ is an isoparametric hypersurface in the total space of $ξ$ equipped with a connection metric, and for $r>0$ small enough, the induced metric of $S_r(ξ)$ has positive Ricci curvature under the additional assumptions that $M$ has a metric with positive Ricci curvature and $n\geq3$. As an application, if $M$ admits a metric …
The rigidity of sharp spectral gap in nonnegatively curved spaces
该研究将非负Ricci曲率下的第一尖锐谱隙刚性推广到紧RCD(0,N)空间，证明谱隙取极值等价于空间为一维常密度，所用的Sobolev理论与奇异一维局部化结合的新技术有独立价值，并导出几乎刚性结果。
We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on the 3-torus. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (Dk) for the fixed points of the involution on the 3-torus and of cyclic type (Ak) otherwise. The collapsing metrics…
On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of $k$ here is optimal, and hence answers an open question of Funano. This approach works still on a compact finite-dimensional Alexandrov space of nonnegative curvature and proves affirmatively a conjecture of Funano and Shioya asserting a dimension free upper bound for eigenvalue ratios i…
A splitting theorem for manifolds with a convex boundary component and applications
研究具有Ricci曲率下界的流形的分裂定理，通过要求边界分量抛物且凸，利用度量胶合及非光滑设置下基于最优传输的综合工具，得出分裂定理及相关刚性结果，回答了特定的刚性问题。
We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lovász-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require in addition an upper bound on the sectional curvature of the space, which permits us to use comparison tools in Cartan-Alexandrov-Toponogov (or CAT) spaces. Along the way, we also quantitatively improve our previous result that weak concentration assumptions imply a Cheeger-type isoperimetric bound, to a sharp bound with respect to all parameters.
In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place. The description of Einstein metrics as the result of a gluing-perturbation procedure also sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we…
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