On the moduli space of positive Ricci curvature metrics on homotopy spheres
David J. Wraith
证明了某些同伦球面上Ricci正度量的模空间有无穷多个连通分支。
We show that the moduli space of Ricci positive metrics on certain homotopy spheres has infinitely many connected components.
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Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature
讨论了奇维同伦球面的Sasaki几何。特别地，我们给出了一个全新的证明存在度量的积极的里奇曲率奇异球，可以实现为边界的平行化流形。此外，还证明了在这样的同伦球面$\scriptstyle{Σ^{2n+1}}$上，Sasakian结构的模空间具有无限多个由不等价的底层切触结构确定的正分量。我们还证明了在$4m+1$维同伦实射影空间的每一个已知的$\scriptstyle{2^{2m}}$不同的同构类型上存在具有正里奇曲率的Sasakian度量.
We show that the moduli space of Riemannian metrics of nonnegative sectional curvature on certain quotients of Brieskorn varieties in dimension 5 has infinitely many path components. The same is true for the corresponding moduli space of positive Ricci curvature.
We examine the moduli spaces of Type~A connections on oriented and unoriented surfaces both with and without torsion in relation to the signature of the associated symmetric Ricci tensor. If the signature of the symmetric Ricci tensor is (1,1) or (0,2), the moduli spaces are smooth. If the signature is (2,0), there is an orbifold singularity.
We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy ${\mathbb {R}} P^5$ has infinitely many path components. We also show that in each dimension $4k+1$ there are at least $2^{2k}$ homotopy ${\mathbb {R}} P^{4k+1}$s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions $4k+3\geq 7$.
We show that the moduli space of Ricci flat metrics of unit volume (including orbifold metrics) on a K3 surface is simply connected and that it has the same rational cohomology as the automorphism group of the K3 lattice $(-E_8)^{\oplus 2}\oplus U^{\oplus 3}$.
We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing $k$-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are. This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro--Wilhelm our approach partly generalizes recent work by Berndt--Olmos o…
In this paper we modify the coordinate construction in our previous paper on the universal moduli space of pair consisting of a Riemann Surfaces and a stable holomorphic bundles on the Riemann Surface, so as to produce a new set of coordinates, which are in fact K\" ahler coordinates on this universal moduli space. Further, we give a functional determinant formula for the Ricci potential of the universal moduli space.
Positive Bakry-Émery Ricci Curvature on Homotopy Spheres
该研究针对Wei和Wylie的问题，在维度8k+1和8k+2中构造出反例，证明同伦球虽存在满足Ric_g + Hess_g f > 0的加权核度量，但因α-不变量非零而无Ric≥0的黎曼度量。
On moduli spaces of Ricci solitons
研究了紧致流形M上收缩Ricci孤子的形变，推广了经典的爱因斯坦度规形变理论。利用M上所有黎曼度量空间中扭曲切片S_f的适当概念，定义了无穷小孤子形变和局域孤子预模空间.本文证明了S_f x C^infty（M）的一个有限维子流形的存在性，该子流形包含固定收缩Ricci孤子周围的孤子的预模空间作为解析子集.我们定义了孤子刚性，并给出了包含它的准则。
Space of Ricci flows (I)
本文研究具有有界能量和数量曲率的非坍缩里奇流解的模空间。我们证明了这种模空间的一个弱紧定理，并应用它研究了等周常数控制、Kähler里奇流和梯度收缩孤子的模空间.
We show that the moduli space of positive Ricci curvature metrics on all the total spaces of $S^7$-bundles over $S^8$ which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an inf…
Moduli spaces of Ricci positive metrics in dimension five
本文利用 spin$^c$ Dirac 算子的 $\eta$ 不变量区分正 Ricci 曲率度量模空间的连通分支，并构造无穷多个五维主 $S^1$ 丛流形使其模空间各有无穷多分支。
Stability and moduli space of generalized Ricci solitons
本文研究了广义里奇流形的稳定性与模空间，通过分析规范变换群和新连接，推导出二阶变分公式，并证明毕许特平坦流形的稳定性及变形特性。
We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give complex analytic charts on the topological space consisting of Kähler-Ricci solitons, by studying differential geometric aspects of this infinite dimensional moment map. Some stacky words and arguments on Gromov-Hausdorff convergence help to glue them together in the holomorphic mann…
Metrics of positive Ricci curvature on vector bundles over nilmanifolds
我们构造了环面上（或更一般地说，零流形上）某些向量丛的正里奇曲率度量.这就产生了第一个例子的流形与积极的里奇曲率同伦等价，但不同胚流形的非负截面曲率。
Ricci-flat deformations and special holonomy
设G是Ricci平坦和乐群SU（n），Sp（n），Spin（7）或G_2中的一个，M分别是2n，4 n，8或7维的紧致流形.我们证明了M上无挠G-结构的模空间到Ricci-平坦度量的模空间的自然映射是开的，并且它的象是一个光滑流形.对于例外情形G = Spin（7）和G_2，我们将结果推广到渐近圆柱流形。
Non-collapsing of Ricci shrinkers with bounded curvature
为具有限第二同伦群和一致曲率界的单连通Ricci收缩子建立一致熵界，并将非塌陷结果扩展至满足Bakry-Émery条件的光滑度量测度空间。
Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach
本文考虑环面$T^{2n}$上Ricci型辛联络的解析曲线$\nabla^t$，其中$\nabla^0$为标准联络.我们表明，由递归参数，如果$\nabla^t$是一个正式的曲线，这样的连接，然后存在一个正式的曲线的辛同构$ψ_t$，使$ψ_t\cdot\nabla^t$是一个正式的曲线平坦不变辛连接，所以$\nabla^t$是平坦的所有$t$。将这个结果应用于解析曲线的泰勒级数，意味着从$\nabla^0$开始的Ricci型辛联络的解析曲线也是平坦的。环面$(T^{2n},ω)$的辛同构群$G$作用在Ricci型辛联络空间$\E$上。作为研究模空间$\E/G$的一个初步工作，我们研究了在辛同胚形式曲线作用下联络形式曲线的模。
We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expanders is connected.
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 …
Homotopy groups of moduli spaces of representations
我们计算了李群GL（n，C）和U（p，q）中紧致定向曲面表示的模空间的某些同伦群.我们的方法依赖于解释这些表示的希格斯束和使用博特-莫尔斯理论上相应的模空间。
We examine moduli spaces of locally homogeneous surfaces of Type~$\mathcal{B}$ with torsion where the symmetric Ricci tensor is non-degenerate. We also determine the space of affine Killing vector fields in this context.
Canonical Metrics on the Moduli Space of Riemann Surfaces I
证明了几种经典完备度量在Teichmüller空间和Riemann曲面的模空间上的等价性。我们利用两个新的Kähler度量：Ricci度量和扰动Ricci度量作为桥梁，证明了扰动Ricci度量是一个完备的Kähler度量，它具有有界的负全纯截面曲率和有界的双截面曲率以及里奇曲率.作为推论，我们证明了这两个新的度量等价于几个著名的经典度量，其中包括Teichmüller度量，因此也包括Kabayashi度量，Kähler-Einstein度量和McMullen度量.这也解决了丘在80年代初的一个猜想。
Homotopy G-algebras and moduli space operad
本文强调了代数曲线模空间在结合代数与拓扑中的作用，证明了带乘法operad空间构成同伦Gerstenhaber代数，奇异上链复形自然构成operad，且装饰模空间operad在Kähler流形的de Rham复形上诱导出最一般的同伦G-代数结构。
Homotopy groups of the moduli space of metrics of positive scalar curvature
我们证明，对于稳定范围内的许多次数，S^n和其他流形上的正数量曲率度量模空间的同伦群是非平凡的。这是通过进一步发展，然后应用一个家庭版本的手术建设的Gromov-Lawson到异国情调的顺利家庭的领域由于Hatcher。如前所述，这适用于所有合适维数的流形，以及正数量曲率度量空间的商，它们是通过固定一个点及其切空间的复同态子群的（自由）作用实现的。我们还构造了正数量曲率的度量空间与满同胚群之商具有非平凡高同伦群的特殊流形。
We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.
Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial-$C^0$-conjecture of Tian.
Width and finite extinction time of Ricci flow
这是一篇简短的文章，为一般非专业观众提供完整的证明。结果是双重的。首先，我们讨论了一个几何不变，我们称之为宽度，一个流形，并显示它如何可以实现为最小2-球的面积之和。例如，当$M$是一个同伦3-球面时，其宽度可以不严格地说是“拉过”$M$所需的最小2-球面的面积。其次，我们利用这一点得出结论，Hamilton的里奇流在任何同伦3-球面上在有限时间内灭绝。我们选择写这篇文章是因为这里给出的结果和想法非常有用，而且似乎引起了广泛的关注。
Positive Intermediate Ricci Curvature on Fibre Bundles
本文证明了浸没度量的典型变分型结果，结合手术技术为高连通流形和 exotic 球面等例子建立了正中间里奇曲率度量的存在性，并将正里奇曲率模空间结果推广至正中间情形。
We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimates along the Ricci flow. It has various applications, including to show the continuous dependence of the Ricci flow with respect to the initial metric in Gromov-Hausdorff topology with Ricci curvature bounded below, and to show the compactness of the moduli of Kähler manifolds with bounded scalar curvature and a rough locally almost Euclidean condition.
This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it depends on the initial total curvature and the initial length, rather than just on the initial total curvature as was asserted before. This change does not affect the application of these results to prove finite-time extinction when the third homotopy group of the manifold is non-tr…
3-symmetric spaces, Ricci solitons, and homogeneous structures
本文完成黎曼3-对称空间的完整分类，重点刻画III型空间的3-对称度量模空间，发现其中唯一的近Kähler膨胀Ricci孤立子，还将该孤立子构造推广至非紧单李群的有效表示，并给出Ambrose–Singer空间等距群的计算方法。
The main focus of the paper is the investigation of moduli space of left invariant pseudoRiemannian metrics on the cotangent bundle of Heisenberg group. Consideration of orbits of the automorphism group naturally acting on the space of the left invariant metrics allows us to use the algebraic approach. However, the geometrical tools, such as classification of hyperbolic plane conics, will often be required. For metrics that we obtain in the classification, we investigate geometrical properties: curvature, Ricci tensor, sectional curvature, holonomy and parallel vector fields. The classificatio…
Noncompact Manifolds with Nonnegative Ricci Curvature
设（M，d）是度量空间.对于0<r<R，且p在M中，设G（p，r，R）是通过考虑所有以p为基础的、其像包含在半径为r的闭球中的回路而得到的群，并且如果它们之间存在包含在半径为R的开球中的同伦，则识别两个回路。本文研究了具有非负里奇曲率的完备开流形G（p，r，R）群的渐近性态.我们还发现了G（p，r，R）群与度量空间无穷远切锥之间的关系，并证明了具有非负里奇曲率和小线性直径增长的完备开流形的无穷远切锥是它自己的泛覆盖.
We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $\mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $\mathcal{B}$ surfaces which are not Type $\mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.
Let $M$ be a topological spherical space form, i.e. a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.
Homotopy Poisson and BV$_\infty$ structures on derived moduli spaces of flat connections
本文证明拟 Poisson/拟 BV 结构等价于特定同伦 Poisson/BV$_\infty$ 结构，并应用于平坦联络模空间。
Let $M$ be a Milnor sphere or, more generally, the total space of a linear $S^3$-bundle over $S^4$ with $H^4(M;\mathbb{Q})=0$. We show that the moduli space of metrics of nonnegative sectional curvature on $M$ has infinitely many path components. The same holds true for the moduli space of metrics of positive Ricci curvature on $M$.
On the rational homotopy type of a moduli space of vector bundles over a curve
研究亏格为$g\geq 2$的紧致连通Riemann曲面$X$上秩为2的稳定向量丛的模空间${\mathcal N}_X$的有理同伦.辛群$Aut(H_1(X,{\mathbb Z}))=Sp(2g,{\mathbb Z})$对有理同伦群$π_n({\mathcal N}_X) \otimes {\mathbb Q}$有自然作用。我们证明，这个行动扩展到行动的$Sp(2g,{\mathbb C})$上$π_n({\mathcal N}_X) \otimes {\mathbb C}$。我们还表明，$π_n({\mathcal N}_X) \otimes {\mathbb C}$是一个非平凡的$Sp(2g,{\mathbb C})$ -表示每个$n\geq 2g-1$。特别地，${\mathcal N}_X$是一个有理双曲空间。在$g=2$的特殊情况下，我们为每个$n$计算出现在$π_n({\mathcal N}_X) \otimes {\mathbb C}$中的前导$Sp(2g,{\mathbb C})$ -表示。
Ricci-flat deformations of asymptotically cylindrical Calabi--Yau manifolds
研究了一类产生于拟投射流形上的渐近圆柱Ricci平坦Kähler度量.利用Calabi-Yau几何和分析以及Kodaira-Kuranishi-Spencer理论，并在N.Koiso关于紧流形的结果的基础上，我们证明了在相当一般的假设下，渐近柱面Ricci平坦Kähler度量的任何“小”渐近柱面Ricci平坦变形都是Kähler的，可能是关于扰动的复结构的。我们还发现这些小的变形的模空间的尺寸。在$2n$ -流形上的渐近圆柱Ricci平坦度量类中，完整约化为SU（n）是一个开条件.
For a given smooth manifold, we consider the moduli space of Riemannian metrics up to isometry and scaling. One can define a preorder on the moduli space by the size of isometry groups. We call a Riemannian metric that attains a maximal element with respect to the preorder a maximal metric. Maximal metrics give nice examples of self-similar solutions for various metric evolution equations such as the Ricci flow. In this paper, we construct many examples of maximal metrics on Euclidean spaces.
We examine the moduli space of oriented locally homogeneous manifolds of Type A which have non-degenerate symmetric Ricci tensor both in the setting of manifolds with torsion and also in the torsion free setting where the dimension is at least 3. These exhibit phenomena that is very different than in the case of surfaces. In dimension 3, we determine all the possible symmetry groups in the torsion free setting.
We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist closed (respectively, open) manifolds for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statem…
Moduli of Parallelogram Tilings and Curve Systems
我们用平行四边形确定了平面周期镶嵌的模空间的拓扑。对于每个这样的平铺，我们通过平铺的区域曲线将组合数据相关联。我们表明，具有相同的组合数据的所有平铺形式在一个合适的欧几里得空间是同伦等价于一个圆的一个开子集。此外，对于任何选择的组合数据，我们构建了一个规范的瓷砖与这些数据。
The Ricci Curvature of Half-flat Manifolds
利用半平坦流形和非紧$G_2$完整流形之间的关系，得到了半平坦流形的Ricci曲率张量和标量的内禀挠类表达式.我们的表达式测试岩泽和更一般的幂零流形。我们还推导出表达式，在语言的Calabi-Yau模空间，扭转类和Ricci曲率的\emph{particular}半平坦的流形，自然产生通过镜像对称的通量紧化。利用这些表达式，我们得到了这些半平坦流形上的第二型弦论的凯勒模空间的一个约束。
Ricci Flat Metrics, Flat Connections and $G_{2}$-Manifolds
受M理论启发，证明了以紧致平坦三维流形T³/K为底的超凯勒ALE四维流形族构成的七维流形上，复化无挠G₂结构模空间与T³/K上平坦G^ℂ_ADE联络的特定模空间等价。
On the structure of noncollapsed Ricci flow limit spaces
本文研究了闭里奇流在熵有界下的模空间弱紧性，并揭示了极限空间的结构，证明正则部分具有里奇流时空结构，奇点集的余维至少为四。
On $7$-manifolds with $b_{2}=2$: diffeomorphism classification and nonconnected moduli spaces of positive Ricci curvature metrics
本文推导了$b_2=2$的一类单连通7维流形的$s$-不变量，据此对特定圆周丛全空间做部分微分同胚分类，证明存在正Ricci曲率度量空间及模空间均有无穷多道路分支的7维流形，并确定了所需的配边群。
Compact Mean Convex Hypersurfaces and the Fundamental Group of Manifolds with Nonnegative Ricci Curvature
证明了在非负里奇曲率的非紧连通完备黎曼n-流形N中存在嵌入的紧致无边界严格正中曲率超曲面S，这意味着由包含所诱导的S和N的基本群之间的同态是满射的，只要N-S有两个连通分支，其中一个分支相对于S具有非紧闭包和平凡同伦.证明的思想是将N看作一个合适的洛伦兹流形中的类空超曲面，然后应用甘农和李关于时空拓扑的某些经典结果的最新版本。作为应用，我们证明了如果N是渐近平坦的，那么它只有一端，并且当n大于3时，N是单连通的。
Moduli Spaces of Positive Curvature Metrics in Dimension Four and Beyond
研究四维及更高维正曲率度量的模空间，构造非平凡同调群元素。
Odd-dimensional manifolds with infinitely many different geometries of positive Ricci curvature
针对n≥5的奇数维闭流形，证明其存在无限多个不同的正Ricci曲率几何，即正Ricci曲率度量模空间有无限多个连通分支，丰富了奇数维流形正Ricci曲率几何的研究。
We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart to a previous result of the authors concerning the existence of positive Ricci curvature on highly connected manifolds in dimensions 4j-1 for j > 1, and in dimensions 4j+1 for j > 0 with torsion-free co…
Fibrewise Orbifold Resolutions with Applications to $\mathrm{G}_2$-Moduli Spaces
通过扭曲爆破族解消定制轨道流形的奇点，构造流形丛，并利用实同伦理论证明这些丛在$\pi_2(B\mathrm{hAut}(M)_0)$中生成自由子群，进而证明广义Kummer构造得到的无挠$\mathrm{G}_2$结构模空间的第二同伦群包含正秩自由子群。
Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds
构造截面曲率有下界、正里奇曲率下界的坍缩序列，极限为不可光滑流形，并研究其拓扑与几何性质。
Synthetic G-jet-structures in modal homotopy type theory
本文在带单子模态的同伦类型论中构造无挠G-射流结构的模叠，适用于配稳定分解系统的∞-拓扑空间，以低复杂度表述并提供形式化，分解系统可由deRham-Stack构造给出。
Kähler-Ricci solitons on Fano threefolds with non-trivial moduli
本研究发现了具有非平凡模的复杂度2$\bbT$-Fano三维流形上的Kähler-Ricci孤立子，建立加权Abban-Zhuang估计，证明加权K-稳定性与GIT稳定性等价，推广Koiso定理到对数Fano情形，得到首批严格加权K-半稳定Fano簇等新例子。
Homotopy Type of the Space of Fibrations of the Three-sphere by Simple Closed Curves
本研究确定了三维球面简单闭曲线光滑纤维化模空间的同伦型，证明其与Hopf纤维化子空间同伦型一致，还刻画了该模空间的Fréchet流形结构与建模空间。
Nonnegative pinching, moduli spaces and bundles with infinitely many souls
证明了在每一维$n\ge 10$中存在无穷多个同伦等价但互不同胚的闭单连通黎曼$n$ -流形序列，这些流形具有$0\le \sec\le 1$、正里奇曲率和一致有界直径.我们还构造了一个固定的非同胚型的开流形，它允许无穷多个完全的非负箍缩度量，这些度量具有有界直径的灵魂，使得这些灵魂相互不同胚。最后，我们构造了具有$\sec\ge 0$的完备度量的模空间具有无穷多个连通分支的非紧流形的例子。
Canonical Metrics on the Moduli Space of Riemann Surfaces II
本文继续研究Riemman曲面的Teichmüller和模空间上的正则度量。我们首先证明了伯格曼度量和Carathéodory度量与Kähler-Einstein度量的等价性，解决了丘的另一个旧猜想。证明了扰动Ricci度量的里奇曲率具有负的上、下界，并且具有有界几何.然后详细研究了Kähler-Einstein度量的边界表现，证明了它具有有界几何，并且其曲率的所有共变导数在Teichmüller空间上一致有界.作为我们对这些度量的详细理解的应用，我们证明了模空间的对数余切丛在Mumford意义下是稳定的。
Energy-minimizing maps from manifolds with nonnegative Ricci curvature
本文证明非负Ricci曲率流形到无共轭点流形的同伦类映射能量有下界，等号当且仅当全测地，并推广至有限覆盖与分裂情形。
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