Hans-Joachim Hein, Song Sun, Jeff Viaclovsky, Ruobing Zhang
We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.
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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 estimate along the Ricci flow. It can be used to show the Gromov-Hausdorff convergence of the Kähler Ricci flow on each minimal projective manifold of general type.
For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers.
We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.
In this note we derive an improved no-local-collapsing theorem of Ricci flow under the scalar curvature bound condition along the worldline of the basepoint. It is a refinement of Perelman's no-local-collapsing theorem.
We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.
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.
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。
We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kahler metrics.
We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of $X$ over its canonical model $X_{can}$. This assumption on the Ricci curvature always holds when the Kodaira dimension of $X$ is $n$, $n-1$ or when the general fibre of $X$ over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to…
Any expanding homogeneous Ricci soliton (in particular any homogeneous Einstein manifold of negative scalar curvature) can be obtained, up to isometry, from a Lie subgroup of a nilpotent Iwasawa group $N$ whose induced metric is a Ricci soliton. By nilpotent Iwasawa group we mean the nilpotent Lie group $N$ of the Iwasawa decomposition associated with a symmetric space of non-compact type. Motivated by this fact, in this paper we classify codimension one Lie subgroups of any nilpotent Iwasawa group $N$ whose induced metric is a Ricci soliton.
We consider Riemannian 4-manifolds that Gromov-Hausdorff converge to a lower dimensional limit space, with the Ricci tensor going to zero. Among other things, we show that if the limit space is two dimensional then under some mild assumptions, the limiting four dimensional geometry away from the curvature blowup region is semiflat Kaehler.
Let $X = M \times E$ where $M$ is an $m$-dimensional Kähler manifold with negative first Chern class and $E$ is an $n$-dimensional complex torus. We obtain $C^\infty$ convergence of the normalized Kähler-Ricci flow on $X$ to a Kähler-Einstein metric on $M$. This strengthens a convergence result of Song-Weinkove and confirms their conjecture.
Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces
该研究证明直纹面上有限时间坍塌Kähler-Ricci流产生以$\text{P}^1\times \text{C}$为模型的I型奇点，并由此得到直纹面纤维的最优坍塌速率。
On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows
本文研究纤维空间上Calabi-Yau度量与Kähler-Ricci流的坍缩极限，在判别轨迹除子部分具简单正规交叉时确定Gromov-Hausdorff极限，并给出奇异集Hausdorff测度显式界及底空间极限度量的双有理几何充分条件。
We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $κ$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries.
The purpose of this paper is to introduce the Ricci Yang-Mills soliton equations on nilpotent Lie groups. In the 2-step nilpotent setting, we show that these equations are strictly weaker than the Ricci soliton equations. Using techniques from Geometric Invariant Theory, we develop a procedure to build many different kinds of Ricci Yang-Mills solitons. We finish this note by producing examples of Lie groups that do not admit Ricci soliton metrics but that do admit Ricci Yang-Mills soliton metrics.
On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces
本文研究 Hirzebruch 曲面上塌缩的 Kähler-Ricci 流的有限时间奇点，证明其切流对应于具有有限多个 orbifold 奇点的非平坦梯度 Kähler-Ricci 收缩孤立子。
In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial $μ$-entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case; espeically, the lower bounds of the renormalized heat kernel, observed on a scale comparable to the initial diameter, matches with the lower bound of the renormalized volume ratio, giving the desired distance distortion estimate.
We study solutions to generalized Ricci flow on four-manifolds with a nilpotent, codimension $1$ symmetry. We show that all such flows are immortal, and satisfy type III curvature and diameter estimates. Using a new kind of monotone energy adapted to this setting, we show that blowdown limits lie in a canonical finite-dimensional family of solutions. The results are new for Ricci flow.
This short note studies the collapsing behavior of the Kähler-Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion X -> B where B is a Kähler manifold of lower dimension than X. We give cohomological and curvature conditions under which the fibers collapse at the optimal rate ~(T-t)^{1/2}
Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups
本文对带乘积结构的洛伦兹四维幂零李群，计算两类联络及其曲率并分类其对应的代数里奇孤子。
Nilpotent Lie algebras obtained by quivers and Ricci solitons
本研究提出一种利用无环有限箭图的路径概念构造幂零李代数的方法，证明对应单连通幂零李群容许左不变里奇孤立子，可构造任意高幂零度的里奇孤立子幂零流形族。
We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of M. Garcia-Fernandez and J. Streets. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows, such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the ge…
Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature
本文通过构造新泛函研究幂零主丛上里奇流 blowdown 极限的刚性，证明三维海森堡群情形下极限为扩张里奇孤子，并结合 Lott 的阿贝尔结果完成四维情形的完整局部分类。
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.
We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first Chern class.
It is known that a closed collapsed Riemannian $n$-manifold $(M,g)$ of bounded Ricci curvature and Reifenberg local covering geometry admits a nilpotent structure in the sense of Cheeger-Fukaya-Gromov with respect to a smoothed metric $g(t)$. We prove that a canonical nilpotent structure over a regular limit space that describes the collapsing of original metric $g$ can be defined and uniquely determined up to a conjugation, and prove that the nilpotent structures arising from nearby metrics $g_ε$ with respect to $g_ε$'s sectional curvature bound are equivalent to the canonical one.
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kahler-Einstein metric from the base. Some of our estimates are new even for the Kahler-Ricci flow. A consequence of our result is that, on every minimal non-Kahler surface of Kodaira dimension one, the Chern-Ricci flow converges in the sense of Gromov-Hausdorff to an orbifold Kah…
The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3 \times {\Bbb R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_3 \times {\Bbb R}$, ~ $g_λ^{+}$ is a shrinking while $g_λ^{+}$ and $g_μ$ are expanding and also $g_0^1, g_0^2, g_0^3$ have Ricci solitons. We exhibit among the non-isometric left invariant Lorentz metric on the group $G_4$ only $g_1^λ, g_2^λ$ have Lorentz Ricci solitons and $g_2^λ$ is a shrinking.
We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to admit a Ricci soliton, but we construct them \textit{explicitly} on each group. In the first case, this is done using Lott's blowdown method, whereby we demonstrate convergence of arbitrary diagonal metrics to the solitons. In the second case, which is more complicated, we obtain the solitons using a suitable ansatz.
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…
Compact Manifolds with Unbounded Nilpotent Fundamental Groups and Positive Ricci Curvature
本文构造了一族紧流形，其基本群不是一致几乎交换的，但具有一致正Ricci曲率，从而回答了Kapovitch和Wilking工作遗留的问题。
In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the deformation of spectrum and length spectrum on compact nilmanifolds obtained of Heisenberg and quaternion nilpotent Lie groups.
Collapsing immortal Kähler-Ricci flows
本文研究带半丰沛典范丛的紧Kähler流形上的Kähler-Ricci流，通过演化度量的渐近展开证明其坍缩为饭高纤维化底的典范度量，验证了Song-Tian的两个猜想。
In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the M_{i} themselves, this structure lives over X and allows for a convenient notion of global convergence as well as the appropriate background structure for doing analysis on X. This structure is new even in the case of uniformly bounded curvature and as an application we give a generali…
The prescribed Ricci curvature problem on 5-dimensional nilpotent Lie groups
本文在5维幂零李群上，利用Milnor型定理技术证明了预定Ricci曲率问题中度量与正常数对的全局存在性。
Nilpotent Lie algebras obtained by ordered sets and Ricci solitons
利用传递反对称有序集（TAOS）的关联代数构造幂零李代数，并证明由特殊TAOS（阵列TAOS）得到的幂零李代数是代数Ricci孤子，推广了有限无环箭图构造代数Ricci孤子的结果。
We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilating about infinite time singularities. In this case we classify the possible Gromov-Hausdorff limits and construct 2-dimensional virtual limits. The virtual limits are constructed using Fukaya theory of the limits of local covers. We then show that the virtual limit arising from a…
We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the $2k+1$-dimensional Heisenberg Lie group $H_{2k+1}$ carries a Ricci flat left invariant Lorentzian metric if and only if $k=1$. We show also that for any $2\leq q\leq k$, $H_{2k+1}$ carries a Ricci flat left invariant pseudo-Riemannian metric of signature $(q,2k+1-q)$ and we give explicite examples of such metrics.
We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter ~ e^{-t/2}. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with …
We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized Kähler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. …
In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some …
The Ricci-flatness that lurks in weight
该研究提出两种基于权重组合性质的构造方法，可在幂零李群上得到左不变里奇平坦伪黎曼度量，并证明了低维幂零李代数存在不定里奇平坦度量。
Semi-local simple connectedness of non-collapsing Ricci limit spaces
本文证明非塌缩Ricci极限空间中任一点附近的小球内回路可在稍大球中收缩，从而得到该空间具有半局部单连通性。
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…
We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat.
We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of the Kahler-Ricci flow on compact Kahler manifolds with semiample canonical bundle and intermediate Kodaira dimension. Our results allow us to understand their collapsed Gromov-Hausdorff limits when the base is smooth and the discriminant has simple normal crossings.
We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))|g(t) converges smoothly to a Ricci soliton, uniformly on compact sets. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group,…
This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not necessarily "maximally degenerating". Our results also give a proof of Kontsevich-Soibelman [KS04, Conjecture 1] (cf., [GW00, Conjecture 6.2]) in the case of K3 surfaces as a byproduct.
Examples of Ricci limit spaces with non-integer Hausdorff dimension
本文构造了首批奇异集Hausdorff维数超过正则集且空间Hausdorff维数为非整数的塌缩Ricci极限空间例子，回答了Cheeger-Colding的相关问题。
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…
This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\leq8$ are obtained. Some related open questions are presented.
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.
We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat Kahler metrics, with an emphasis on the case when the metrics are volume collapsing.
Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannian metric, $\mathfrak{g}$ its Euclidean Lie algebra and $Z(\mathfrak{g})$ the center of $\mathfrak{g}$. By using an orthonormal basis adapted to the splitting $\mathfrak{g}=(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])\oplus O^+\oplus (Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp)\oplus O^-$, where $O^+$ (resp. $O^-$) is the orthogonal of $Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]$ in $[\mathfrak{g},\mathfrak{g}]$ (resp. is the orthogonal of $Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]^\perp$ in $[\ma…
We establish a general "boundedness implies convergence" principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows on torus fibered minimal models to obtain convergence results.
We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.
Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature. It has been conjectured by Lauret-Will that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimension 5, as well as for Heisenberg and standard filiform Lie algebras.
We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so that it is fibered by Calabi-Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and of a flat metric on Euclidean space. An analogous result holds for c…
Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the invariant Ricci flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish geome…
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