Igor Belegradek, Guofang Wei
We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E and R^p admits a complete metric of positive Ricci curvature for all large p.
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In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal $G$-bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-Émery Ricci flow as well as Ricci flow on circle bundles over Kähler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.
For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.
We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.
In this paper, we propose a method of studying the modified Kahler-Ricci flow on projective bundles and give the explicit equation from the view point of symplectic geometry.
We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature.
The Ricci flow for circle bundles over surfaces
本文研究曲面上圆周丛的归一化里奇流方程，求解该方程并分析解的渐近行为及其与模型几何的联系。
It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.
We show the existence and uniqueness of a one-parameter family of smooth complete $U(1)$-invariant gradient steady Ricci solitons on the total space of any complex line bundle over a Fano Kähler-Einstein base with first Chern class proportional to that of the base. These solitons are non-Kähler except on the total space of the canonical bundle.
The aim of this paper is to present a direct and simple proof of a result concerning the existence of metrics of positive Ricci curvature on the total space of fiber bundles with compact structure groups. In particular, it also generalizes and puts in a unified framework the results in Nash \cite{nash} and Poor \cite{poor}. With the intention of disseminating this result, we apply it to build new examples of manifolds with positive Ricci curvature, including bundles whose base consists of gradient shrinking Ricci solitons.
We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.
We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base.
Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on a flat Riemannian manifold. …
We give an explicit description of all complete $G$-invariant Ricci-flat Kähler metrics on the tangent bundle $T(G/K)\cong G^\bbC/K^\bbC$ of rank-one Riemannian symmetric spaces $G/K$ of compact type, in terms of associated vector-functions.
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.
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。
For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.
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 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。
In this paper, we prove that expanding gradient Ricci solitons with (positively) pinched Ricci curvature are trivial ones. Namely, they are either compact or flat.
We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $κ$-collapsed and $κ$-noncollapsed ancient solutions of type I are obtained on circle bundles for all odd dimensions $\geq 7$. In dimension $7$ such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds. Continuous families of $κ$-collapsed ancient solut…
Infinitely many non-collapsed steady Ricci solitons on complex line bundles
研究在复射影空间上的复线丛O(k)上构造无限多个非收缩的里奇溶剂，并证明其具有渐近锥形的里奇平坦度量。
We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3d complete Riemannian manifold with non-negative Ricci curvature, there exists a smooth Ricci flow starting from it.
We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton structures on the vector bundles whose distance sphere bundles are respectively the twistor and ${\rm Sp}(1)$ bundles over quaternionic projective space.
We show that in dimension 4 and above, the lifespan of Ricci flows depends on the relative smallness of the Ricci curvature compared to the Riemann curvature on the initial manifold. We can generalize this lifespan estimate to the local Ricci flow, using which we prove the short-time existence of Ricci flow solutions on noncompact Riemannian manifolds with at most quadratic curvature growth, whose Ricci curvature and its first two derivatives are sufficiently small in regions where the Riemann curvature is large. Those Ricci flow solutions may have unbounded curvature. Moreover, our method imp…
In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature should be Ricci flat. The result answers a question in case of Kähler-Ricci solitons proposed by Chow, Lu and Ni in a book.
The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics.
In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.
If $π:M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature.
We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that it is a quotient of $Σ\times \mathbb{C}^{n-1-k}\times N^k$, where $Σ$ and $N$ denote the Hamilton's cigar soliton and some compact Kähler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kähler Ricci soliton with $Ric\geq 0$, together with subquadratic volume growth or $\limsup_{r\to \infty} rS<1$ must have u…
Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructed from a gradient shrinking Ricci soliton.
We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\leq Ct^{-1}$ for all $t\in (0,T]$, where $C$ depends only on the dimension $n$ and $T$ depends on $K$ and the injectivity radius $inj_{g(t)}$. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly…
In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.
In this paper, we study the injectivity radius bound for 3-d Ricci flow. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 of the book of Chow-Lu-Ni (2006).
We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.
We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all satisfied if the manifold is curvature homogeneous. This leads to a complete description of four- and six-dimensional Kaehler gradient Ricci solitons with constant scalar curvature.
In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $κ$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.
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.
Expanding Ricci solitons and Higgs bundles
本文研究了具有幂零对称性的扩展里齐流形与Higgs丛的关系，通过建立对应关系产生新的非局部同构例子，并在四维情况下给出完整描述。
In this paper, we prove that any $κ$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $ε$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $κ$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies $\lim_{r(x)\rightarrow\infty}R(x)f(x)=C_0\sup_{x\in M}R(x)$ with $C_0>\frac{n-2}{2}$.
We give a description of all $G$-invariant Ricci-flat Kähler metrics on the canonical complexification of any compact Riemannian symmetric space $G/K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G/K)$. As the simplest application, we obtain the explicit description of the set of all complete $\mathrm{SO}(3)$-invariant Ricci-flat Kähler metrics on $T{\mathbb S}^2$, which includes the well-known Eguchi-Hanson-Stenzel metrics and a new one-parameter family of metrics.
We study the behavior of the Kähler-Ricci flow on some Fano bundle which is a trivial bundle on one Zariski open set. We show that if the fiber is $\mathbb{P}^{m}$ blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable kähler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov-Hausdorff sense to a metric on the base.
In this short note we discuss some recent results about two-positive Ricci curvature and their applications to positive Einstein curvature.
This paper is a survey on the structure of manifolds with a lower Ricci curvature bound.
We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.
We discuss an elementary consequence of the works of (1) Brett Kotschwar and Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang.
We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.
Ancient Ricci flows with nonnegative Ricci curvature
本文研究了非负曲率非塌陷古老里奇流的渐近几何，通过其无穷远处的非塌陷F-极限度量孤子得出两个分歧定理。
We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset.
Riemannian metrics of positive Ricci curvature were constructed on certain moment-angle manifolds.
We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces.
In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.
We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.
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…
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.
In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.
Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {\em almost nonpositive $k$-Ricci curvature}, which is weaker than the existence of a Kähler metric with nonpositive $k$-Ricci curvature. When $k=1$, this is just the {\em almost nonpositive holomorphic sectional curvature} introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with $k$-Ricci curvature bounded from above by a pos…
We use a first-order energy quantity to prove a strengthened statement of uniqueness for the Ricci flow. One consequence of this statement is that if a complete solution on a noncompact manifold has uniformly bounded Ricci curvature, then its sectional curvature will remain bounded for a short time if it is bounded initially. In other words, the Weyl curvature tensor of a complete solution to the Ricci flow cannot become unbounded instantaneously if the Ricci curvature remains bounded.
B. Y. Chen establish the relationship between the Ricci curvature and the squared mean curvature for submanifolds of Riemannian space form with arbitrary codimension. In this paper, we generalize the relationship between the Ricci curvature and the squared norm of mean curvature vector for submanifolds of Bochner Kahler manifolds.
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.
We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms and C-totally real submanifolds of Sasakian space forms.
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