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 $M$, $vol(M)\geq v>0$, then $M$ is diffeomorphic and Gromov-Hausdorff close to a hyperbolic manifold where $δ, ε$ also depends on $v$.
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.
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 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.
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 prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.
In this paper, we prove some rigidity results for both shrinking and expanding Ricci solitons. First, we prove that compact shrinking Ricci solitons are Einstein if we control the maximum value of the potential function. Then, we prove some rigidity results for non-compact gradient expanding and shrinking Ricci solitons with pinched Ricci and scalar curvatures, assuming an asymptotic condition on the scalar curvature at infinity.
We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.
We introduce a new curvature-pinching condition, which is weaker than the positive sectional curvature or PIC1, and then we prove several rigidity results for the rotationally symmetric solutions of steady Ricci solitons, i.e., the Bryant Ricci solitons.
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 will show the Cheeger-Colding segment inequality for manifolds with integral Ricci curvature bound. By using this segment inequality, the almost rigidity structure results for integral Ricci curvature will be derived by a similar method as in \cite{CC1}. And the sharp Hölder continuity result of \cite{CoN} holds in the limit space of manifolds with integral Ricci curvature bound.
Let $M$ be an open Riemannian $n$-manifold with nonnegative Ricci curvature. We prove that if the first Betti number of $M$ equals $n-1$, then $M$ is flat.
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…
Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.
We define a gradient Ricci soliton to be rigid if it is a flat bundle $% N\times_Γ\mathbb{R}^{k}$ where $N$ is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.
We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity results for the smallest Dirichlet eigenvalues for the weighted p-Laplacians.
A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Qing}.
We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on $\mathbb{R^n}$ , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, we extend known $L^p$ bounds and decay rates for Ricci DeTurck flow and prove regularity of the flow at the initial data.
In this paper we introduce the notion of rigidity for harmonic-Ricci solitons and we provide some characterizations of rigidity, generalizing some known results for Ricci solitons. In the compact case we are able to deal with not necessarily gradient solitons while, in the complete non-compact case, we restrict our attention to steady and shrinking gradient solitons. We show that the rigidity can be traced back to the vanishing of certain modified curvature tensors that take into account the geometry a Riemannian manifold equipped with a smooth map $φ$, called $φ$-curvatures, which are a natur…
We prove that an $n$-dimensional, $n\geq4$, compact gradient shrinking Ricci soliton satisfying a $L^{\frac n2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^n$, which improves the rigidity theorem given by G. Catino (arXiv:1509.07416vl).
We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity theorems for the smallest Dirichlet eigenvalues for the weighted $p$-Laplacians.
In this paper, we prove that the round cylinders are rigid in the space of Ricci shrinkers. Namely, any Ricci shrinker that is sufficiently close to $S^{n-1}\times \mathbb R$ in the pointed-Gromov-Hausdorff topology must itself be isometric to $S^{n-1}\times \mathbb R$.
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties.
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…
In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on $\mathbb{S}^2\times \mathbb{R}^2$ (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.
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.
In this paper, we prove that complete gradient steady Kähler-Ricci solitons with harmonic Bochner tensor are necessarily Kähler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) Kähler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times \mathbb{C}^{n-k}$, where $N$ is a Kähler-Einstein manifold with positive (or negative) scalar curvature.
We show that $S^2\times S^2$ is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for $S^2\times N$, where $N$ belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.
Transverse Rigidity of Shrinking Sasaki-Ricci Solitons
National Taiwan University（台湾大学）；Shanghai Institute of Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Chongqing University of Technology（重庆理工大学）；National Taiwan Normal University（台湾师范大学）；Wuhan University（武汉大学）
In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.
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 classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics.
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.
Lu-Minguzzi-Ohta have introduced the notion of a lower $N$-weighted Ricci curvature bound with $\varepsilon$-range, and derived several comparison geometric estimates from a Laplacian comparison theorem for weighted Laplacian. The aim of this paper is to investigate various rigidity phenomena for the equality case of their comparison geometric results. We will obtain rigidity results concerning the Laplacian comparison theorem, diameter comparisons, and volume comparisons. We also generalize their works for non-symmetric Laplacian induced from vector field potential.
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.
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}
We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\mathbb{R}^4$, $\mathbb{R}^2\times\mathbb{S}^2$ or the round cylinder $\mathbb{R}\times\mathbb{S}^3$. Under the condition of fourth order divergence-free Weyl tensor, we have the same results.
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. …
