Takashi Shioya, Takao Yamaguchi
In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.
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We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of nonnegative sectional curvature or satisfies Thurston's Geometrization Conjecture. We also prove a version with boundary. Kleiner and Lott independently proved similar results.
Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.
Let M be a closed 5-manifold of pinched curvature 0<δ\le \text{sec}_M\le 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) δ=1/4 and the fundamental group is a non-cyclic group of order at least C, a constant. (ii) The center of the fundamental group has index at least w(δ), a constant depending on δ. (iii) The ratio of the volume and the maximal injectivity radius is less than ε(δ). (iv) The volume is less than ε(δ) and the fundamental group π_1(M) has a center of index at least w, a universal constant, and π_1(M) is either isomor…
We prove that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. This theorem was stated by Perelman and was used in his proof of the geometrization conjecture.
We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data satisfying the aformentioned bounds.
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.
In the last two decades, one of the most important developments in Riemannian geometry is the collapsing theory of Cheeger-Fukaya-Gromov. A Riemannian manifold is called (sufficiently) collapsed if its dimension looks smaller than its actual dimension while its sectional curvature remains bounded (say a very thin flat torus looks like a circle in a bared eyes). We will survey the development of collapsing theory and its applications to Riemannian geometry since 1990. The common starting point for all of these is the existence of a singular fibration structure on collapsed manifolds. However, n…
Inradius collapsed manifolds with a lower Ricci curvature bound
本文研究了内径坍缩流形的几何性质，证明其极限空间具有特定的对称性和曲率约束，且边界成分数有限。
In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.
Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature
研究具有双边有界Ricci曲率的非坍缩流形的正交标架丛在几乎标准度量下的Gromov-Hausdorff极限空间，证明其与Ricci极限空间类似，奇异集余维数≥4且补集为开稠密的C^{1,α}黎曼流形。
The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a $n$-manifold of bounded sectional curvature are $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $n$), then up to a diffeomorphism, the underlying nilpotent Killing structures coincide with each other or one is embedded into another as a subsheaf. It improves Cheeger-Fukaya-Gromov's locally compatibility of pure nilpotent Killing structures for one collapsed metric of bounded sectional curvature to …
Topology of two-dimensional collapsed spaces with lower Ricci bounds
证明了Ricci曲率有下界且本质维数为二的塌缩影测度空间是拓扑曲面（可能带边界）。
Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces
本文在Ricci曲率有下界的闭流形及其非坍缩Ricci极限空间上，建立了图拉普拉斯算子特征值与特征函数的高概率谱收敛界，为带奇异点的加权拉普拉斯提供谱逼近。
We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It reveals new, previously unknown, properties that all generalized spaces with a lower Ricci curvature bound must have and it has a number of applications. This new kind of estimate asserts that the geometry of small balls along any minimizing geodesic changes in a Hölder continuous way with a constant depending on the lower bound for the Ricci curvature, the dim…
In this paper we discuss and prove $ε$-regularity theorems for Einstein manifolds $(M^n,g)$, and more generally manifolds with just bounded Ricci curvature, in the collapsed setting. A key tool in the regularity theory of noncollapsed Einstein manifolds is the following: If $x\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ is sufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\ell},y^*)\subset \mathbb{R}^{n-\ell}\times C(Y^{\ell-1})$ for $\ell\leq 3$, then in fact $|Rm|\leq 1$ on $B_1(x)$. No such results are known in the collapsed setting, and in fact it is easy to see wi…
We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme. Given such initial data (M,g) we show that a Ricci flow exists for a short time interval. This enables us to construct a Ricci flow of any (possibly singular) metric space (X,d) which arises as a Gromov-Hausdorff limit of a sequence of 3-manifolds which satisfy a), b) and c) uniformly. As a corollary we show that such an X must be a manifold.
In this paper, we study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds with boundary. We determine the limit spaces of inradius collapsed manifolds as Alexandrov spaces with curvature uniformly bounded below. When the limit space has co-dimension one, we completely determined the topology of inradius collapsed manifold in terms of singular $I$-bundle…
On the local topology of non-collapsed Ricci bounded limit spaces
本文证明了非塌缩Ricci有界黎曼流形的Gromov-Hausdorff极限中，正则部分的局部第一Betti数消失，并讨论了相关应用与未解决问题。
We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curvature bounds.
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。
In this paper we consider closed non-collapsed ancient solutions to the mean curvature flow ($n \ge 2$) which are uniformly two-convex. We prove that any two such ancient solutions are the same up to translations and scaling. In particular, they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution constructed by Brian White in (2000), and by Robert Haslhofer and Or Hershkovits in (2016).
Ricci Flow under Kato-type curvature lower bound
本文将非塌缩Ricci流的存在性理论从逐点曲率下界推广到Kato型下界，证明三维非塌缩强Kato极限空间同胚于光滑流形，并在高维1-迷向曲率条件下成立，同时利用Ricci流光滑化研究数量曲率几何的稳定性问题。
Instability of the fundamental group for non-collapsed Ricci-limits
构造两个闭四维流形序列，具有非负 Ricci 曲率、有界直径和体积，但基本群不同而 Gromov-Hausdorff 极限相同，否定回答了 Pan 的问题。
Collapsed limits of compact Heisenberg manifolds with sub-Riemannian metrics
研究紧Heisenberg流形在次黎曼度量下的坍缩极限，证明其Gromov–Hausdorff极限必为平坦环面，并在附录建立相关Systolic不等式。
In this paper, we construct local and global solutions to the Kähler-Ricci flow from a non-collapsed Kähler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kähler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete Kähler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and m…
Mahler type theorem and non-collapsed limits of sub-Riemannian compact Heisenberg manifolds
本文证明带次黎曼度量的紧Heisenberg流形在直径与Popp测度界下具有非坍缩GH子列极限，且极限仍为同维紧Heisenberg流形。
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.
This is an expositiry article on collapsing theory in Riemannian geometry written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applications of collapsing theory to Riemannian geometry are NOT discussed in this survey.
For a sequence of extrinsic or intrinsic biharmonic maps $u_j: M_j\rightarrow N$ from a sequence of non-collapsed degenerating closed Einstein 4-manifolds $(M_j,g_j)$ with bounded Einstein constants, bounded diameters and bounded $L^2$ curvature energy into a compact Riemannian manifold $(N,h)$ with uniformly bounded biharmonic energy, we establish a compactness theory modular finitely many bubbles, which are finite energy biharmonic maps from $\mathbb{R}^4$, or from $\mathbb{R}^4 / Γ$ for some nontrivial finite group $Γ\subset SO(4)$, or from some complete, noncompact, Ricci flat, non-flat AL…
For $ρ, v>0$, we say that an $n$-manifold $M$ satisfies local $(ρ,v)$-bound Ricci covering geometry, if Ricci curvature $\text{Ric}_M\ge -(n-1)$, and for all $x\in M$, $\text{vol}(B_ρ(\tilde x))\ge v>0$, where $\tilde x$ is an inverse image of $x$ on the (local) Riemannian universal cover of the $ρ$-ball at $x$. In this paper, we extend the nilpotent fiber bundle theorem of Cheeger-Fukaya-Gromov on a collapsed $n$-manifold $M$ of bounded sectional curvature to $M$ of a local $(ρ,v)$-bound Ricci covering geometry, and $M$ is close to a non-collapsed Riemannian manifold of lower dimension. The n…
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 develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.
Rigidity and non-existence results for collapsed translators
本文证明了在R^3中宽度为π且从下界有界的自翻译2表面唯一为grim reaper圆柱体，并利用加权抛物性方法证明了其刚性结果。
In a remarkable article published in 1982, M. Gromov introduced the concept of minimal volume, namely, the minimal volume of a manifold $M^n$ is defined to be the greatest lower bound of the total volumes of $M^n$ with respect to complete Riemannian metrics whose sectional curvature is bounded above in absolute value by 1. While the minimal curvature, introduced by G. Yun in 1996, is the smallest pinching of the sectional curvature among metrics of volume 1. The goal of this article is to provide estimates to minimal volume and minimal curvature on 4-dimensional compact manifolds involving som…
We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $-C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other in…
PIC1 pinched manifolds are flat or compact
本文证明了所有维度下，缩陷里奇曲率的流形要么平坦要么紧致，发展了处理无限远坍缩流形的提升技术，并改进了里奇流的曲率估计和间隙定理。
We extend Cheeger-Gromov's and Anderson's convergence theorems to regular limit spaces of manifolds with bounded Ricci curvature and local covering geometry, by establishing the $C^{1,α}$-regularities that are the best one may expect on those Ricci-limit spaces. As an application we prove an optimal generalization of Fukaya's fibration theorem on collapsed manifolds with bounded Ricci curvature, which also improves the original version to $C^{1,α}$ limit spaces.
On the dimension of the singular set of perimeter minimizers in spaces with a two-sided Ricci curvature bound
本文研究非塌缩 Ricci 极限空间中周长极小元的奇异集，证明其 Hausdorff 维数不超过环境维数减五，并表明该界尖锐。
We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of $L^2$-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of $1$-balls in the universal cover.
We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum of the L2-norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results f…
In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite topological type under some restrictions shown below.
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.
We show that every closed nonpositively curved manifold with non-trivial volume flux group has zero minimal volume, and admits a finite covering with circle actions whose orbits are homologically essential. This proves a conjecture of Kedra-Kotschick-Morita for this class of manifolds.
Scalar curvature, volume, and Weil-Petersson translation length
本文在伪阿诺索夫映射环面上构造满足特定条件的度量，改进了相关不等式的常数，给出了伪阿诺索夫映射环面双曲体积与Weil-Petersson平移长度关系的新证明。
On ends of finite-volume noncompact manifolds of nonpositive curvature
本文证实了一个猜想：在截面曲率介于-1和0之间的有限体积完备非紧流形上，若万有覆盖是能见流形，则每个端的根本群几乎幂零。
A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone
本文证明了对任意在 $\mathbb{S}^3$ 上自由作用的有限子群 $Γ$，存在具有非负 Ricci 曲率且渐近锥为 $C(\mathbb{S}_δ^3 /Γ)$ 的 4 维完备 Riemannian 流形，从而回答了 4 维非塌缩切锥的拓扑障碍问题并实现了其拓扑分类。
We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.
In this work, we (partially) generalize two classical tools in study of collapsed manifolds with bounded sectional curvature: a (singular) fibration theorem by Fukaya (1987) and Cheeger-Fukaya-Gromov (1992), and the stability for isometric compact Lie group actions on manifolds by Palais (1961) and Grove-Karcher (1973), to manifolds with local bounded Ricci covering geometry. Our two generalized results have been used in a recent work of Xiaochun Rong in generalizing Gromov's almost flat manifolds theorem to maximally collapsed manifolds with local bounded Ricci covering geometry.
Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
该文研究N维${\rm RCD}(K,N)$空间的等周比较定理与渐近等周性质，得到几乎正则性定理及函数不等式推论，多数结论在光滑非紧流形经典框架下也为新结果，合成理论起关键作用。
In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize ${\rm CD}(K, \infty)$ condition and non-collapsed ${\rm CD}(K, N)$ condition on Riemannian manifolds with boundary. In particular, using $L^1$-optimal transportation theory, we prove that ${\rm CD}(K, \infty)$ condition implies geodesical convexity.
The Kummer Construction of Calabi-Yau and Hyper-Kähler Metrics on the $K3$ Surface, and Large Families of Volume Non-collapsed Limiting Compact Hyper-Kähler Orbifolds
本文基于奇异摄动与加权函数空间分析，给出佩奇等关于K3曲面里奇平坦凯勒度量库默构造猜想的两个新严格证明，得到改进估计、完整58维模空间及体积非塌缩极限轨形大家族。
We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.
We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and give an application on the algebraic entropy of its fundamental group. We also extend the almost minimal volume rigidity of \cite{BBCG} to integral Ricci curvature.
We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson-Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kahler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, a…
We show two sphere theorems for the Riemannian manifolds with scalar curvature bounded below and the non-collapsed $\mathrm{RCD}(n-1,n)$ spaces with mean distance close to $\fracπ{2}$.
We show that closed manifolds supporting a nonpositively curved metric with negative $([\frac{n}{4}]+1)$-Ricci curvature, have positive simplicial volume. This answers a special case of a conjecture of Gromov.
We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow and volume preserving mean curvature flow.
We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.
On an invariant curvature cone along 4-dimensional Ricci flow
本文研究满足曲率锥约束的四维非紧致完备流形，证明了拓扑与几何间隙定理，并研究了具有曲率下界流形的正则性结果。
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