The intent of this short note is to provide context for and an independent proof of the discovery of Klaus Kroencke that complex projective space with its canonical Fubini--Study metric is dynamically unstable under Ricci flow in all complex dimensions N>1. The unstable perturbation is not Kaehler. This provides a counterexample to a well known conjecture widely attributed to Hamilton. Moreover, it shows that the expected stability of the subspace of Kaehler metrics under Ricci flow, another conjecture believed by several experts, needs to be interpreted in a more nuanced way than some may have expected.
In this short article, we improve the dynamical stability and instability results for Ricci-flat metrics under Ricci flow proved by Sesum and Haslhofer, getting rid of the integrability assumption.
We prove the existence of steady Kähler-Ricci solitons on equivariant crepant resolutions of $\mathbb{C}^n/G$, where $G$ is a finite subgroup of $SU(n)$.
We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If $g$ is not a local maximum of the shrinker entropy, we show that there exists a nontrivial normalized Ricci flow emerging from it. These theorems are analogues of results in the Ricci-flat and in the Einstein case.
Consider the standard action of $SO(2)\times SO(3)$ on $\mathbb{R}^5=\mathbb{R}^2\oplus \mathbb{R}^3$. We establish the existence of a uniform constant $\mathcal{C}>0$ so that any $SO(2)\times SO(3)$-invariant Ricci soliton on $\mathbb{S}^4\subset \mathbb{R}^5$ with Einstein constant $1$ must have Riemann curvature and volume bounded by $\mathcal{C}$, and injectivity radius bounded below by $\frac{1}{\mathcal{C}}$. This observation, coupled with basic numerics, gives strong evidence to suggest that the only $SO(2)\times SO(3)$-invariant Ricci solitons on $\mathbb{S}^4$ are round. We also encou…
In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $div^4Rm^\pm=0$, we show that it is either Einstein or a finite quotient of $\mathbb{R}^4$, $\mathbb{S}^2\times\mathbb{R}^2$ or $\mathbb{S}^3\times\mathbb{R}$. The same result can be obtained under the condition of $div^4W^\pm=0$. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature…
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 prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local minimizers for the same class of functionals.
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.
This article is a continuation of [CMZ21b], where we proved that a Ricci flow with a closed and smooth tangent flow has unique tangent flow, and its corresponding forward or backward modified Ricci flow converges in the rate of $t^{-β}$ for some $β>0$. In this article, we calculate the corresponding $\mathbb{F}$-convergence rate: after being scaled by a factor $λ>0$, a Ricci flow with closed and smooth tangent flow is $|\log λ|^{-θ}$ close to its tangent flow in the $\mathbb{F}$-sense, where $θ$ is a positive number, $λ\gg 1$ in the blow-up case, and $λ\ll 1$ in the blow-down case.
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
In this paper we generalize the neck-stability theorem of Kleiner-Lott to a special class of four-dimensional nonnegatively curved Type I $κ$-solutions, namely, those whose asymptotic shrinkers are the standard cylinder $\mathbb{S}^3\times\mathbb{R}$. We use this stability result to prove a rigidity theorem: if a four-dimensional Type I $κ$-solution with nonnegative curvature operator has the standard cylinder $\mathbb{S}^3\times\mathbb{R}$ as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric.
In this short note, we will prove a volume stability theorem which says that if an n-dimensional toric manifold $M$ admits a $\mathbb{T}^n$ invariant Kähler metric $ω$ with Ricci curvature no less than 1 and its volume is close to the volume of $\mathbb{CP}^n$, $M$ is bi-holomorphic to $\mathbb{CP}^n$.
We prove that a four-dimensional gradient shrinking Ricci soliton with $δW^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional cscK gradient Ricci soliton is either Kähler-Einstein, or a finite quotient of $M\times\mathbb{C}$, where $M$ is a Riemann surface. The main arguments are curvature decompositions, the Weitzenböck formula for half Weyl curvature, and the maximum principle.
In this note, we show that on Hopf manifold $\mathbb S^{2n-1}\times \mathbb S^1$, the non-negativity of the holomorphic bisectional curvature is not preserved along the Chern-Ricci flow.
In this note we prove the following result: Let $X$ be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry and with no essential incompressible space form. Then $X$ is diffeomorphic to $\mathbb{S}^4$, or $\mathbb{RP}^4$, or $\mathbb{S}^3\times \mathbb{S}^1$, or $\mathbb{S}^3\widetilde{\times} \mathbb{S}^1$, or a possibly infinite connected sum of them. This extends work of Hamilton and Chen-Zhu to the noncompact case. The proof uses Ricci flow with surgery on complete 4-manifolds, and is inspired by recent work of Bessi$\grave{e}$res, Besson and …
There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of generalized Tanaka-Webster Reeb parallel Ricci tensor for $M$ in $G_2({\mathbb C}^{m+2})$. By using such parallel conditions, we give complete classifications of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.
We study the Ricci flow on $\mathbb{R}^{n+1}$, with $n\geq 2$, starting at some complete bounded curvature rotationally symmetric metric $g_{0}$. We first focus on the case where $(\mathbb{R}^{n+1},g_{0})$ does not contain minimal hyperspheres; we prove that if $g_{0}$ is asymptotic to a cylinder then the solution develops a Type-II singularity and converges to the Bryant soliton, while if the curvature of $g_{0}$ decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotica…
Let $D$ be a compact Kähler manifold with trivial canonical bundle and $Γ$ be a finite cyclical group of order $m$ acting on $\mathbb{C} \times D$ by biholomorphisms, where the action on the first factor is generated by rotation of angle $2π/m$. Furthermore, suppose that $Ω_D$ is a trivialisation of the canonical bundle such that $Γ$ preserves the holomorphic form $dz \wedge Ω_D$ on $\mathbb C \times D$, with $z$ denoting the coordinate on $\mathbb{C}$. The main result of this article is the construction of new examples of gradient steady Kähler-Ricci solitons on certain crepant resolutions of…
A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $KΔK+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end of the 19th century Ricci-Curbastro has proved that conversely, every point $x$ of a Ricci surface has a neighborhood which embeds isometrically in $\mathbb{R}^3$ as a minimal surface, provided $K(x)<0$. We prove this result in full generality by showing that Ricci surfaces can be locally isometrically embedded either minimally in $\mathbb{R}^3$ or maximally in $\m…
In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to $\mathbb{S}^4$ or $\mathbb{RP}^4$ or $\mathbb{CP}^2$. We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.
We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete $G_2$ metric discovered in [BS89][GPP90]. It lies in the interior of the 1-parameter family on $\bigwedge_-^2\mathbb{CP}^2$. All the Ricci-flat metrics constructed have asymptotically conical…
We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's $λ$-functional adapted to the ALE situation and denoted $λ_{\operatorname{ALE}}$. This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.
We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kähler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso--Cao soliton and the Lü--Page--Pope quasi-Einstein metrics on $\mathbb{CP}^{2}\sharp\overline{\mathbb{CP}}^{2}$ (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang--Zhu soliton on $\mathbb{CP}^{2}\sharp 2\overline{\mathbb{CP}}^{2}$ (here the metric is not known explicitly). Finally,…
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 paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product $\mathbb{R}\times N^{n-1}$, or globally conformally equivalent to the Euclidean space $\mathbb{R}^{n}$ or to the round sphere $\mathbb{S}^{n}$. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.
The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,μ)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,μ)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space…
We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat $\mathbb{R}^4 /\mathbb{Z}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$, and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R}…
We prove the existence of a unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at one point. This completes the classification of such solitons in two complex dimensions.
We extend the sphere theorem of \cite{CGY03} to give a conformally invariant characterization of $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $β(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \cite{CGY03} that if $0 \leq β(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq β(M^4,[g]) < 4(1 + ε)$ for $ε> 0$ small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to…
Let $π:\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))\to \mathbb{P}^{n-1}$ be a projective bundle over $\mathbb{P}^{n-1}$ with $1\leq k \leq n-1$. In this paper, we show that lens space $L(k\, ;1)(r)$ with radius $r$ embedded in $\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))$ is a self-similar solution, where $\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))$ is endowed with the $U(n)$-invariant gradient shrinking Ricci soliton structure. We also prove that there exists a pair of critical radii $r_{1}<r_{2}$ which satisfies the following. The lens space $L(k\, ;1)(r)$ is a self-shrinker if $…
In this paper, we have introduced a new notion of generalized Tanaka-Webster Reeb recurrent Ricci tensor in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Next, we give a non-existence property for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with such a condition.
We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $\mathbb{S}^{d_1+1}\times \mathbb{S}^{d_2}$, where $d_i \geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show that $T$ can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.
We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.
In this article, we get classification theorems for a Ricci soliton on the pseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ taking the potential vector field as the tangent component of the position vector of the pseudo-Riemannian hypersurface, denoted by $(M,g,{\bf x}^T,λ)$ in both Riemannian and Lorentzian settings. First, we obtain the necessary and sufficient condition that a pseudo-Riemannian hypersurface $(M,g)$ in $\mathbb{E}^4_1$ admits a Ricci soliton $(M,g,{\bf x}^T,λ)$. In each of the form of the shape operator of a pseudo-Riemannian hypersurface, we obtain cha…
We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy $|Rm|\leq cR$ for some positive constant $c$. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for $4$-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a $3$-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. …
Using center manifolds and topological degree theory, we construct a new family of complete, $SU(2)$-invariant and steady gradient Ricci solitons on the four-dimensional non-compact cohomogeneity one manifold with group diagram $\mathbb{Z}_4\subset U(1)\subset SU(2)$. We also provide simpler constructions of the existing $U(2)$-invariant steady and complete gradient solitons on the cohomogeneity one manifolds with group diagrams $\mathbb{Z}_n\subset U(1)\subset SU(2)$ for any $n\in \mathbb{N}$, including Appleton's non-collapsed solitons for $n\ge 3$.
We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with $b^+=1$. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topological 4-manifolds ${\mathbb C}{P}^2 # l \overline{{\mathbb C}{P}^2}$, where $5 \leq l \leq 8$.
We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic $\mathbb{S}^{5}$ in the nearly K{ä}hler sphere $\mathbb{S}^6$. Being locally isometric to a pseudoholomorphic curve in $\mathbb{S}^5$ turns out to be equivalent to the Ricci-like condition $Δ\log(1-K)=6K,$ where $K$ is the Gaussian curvature of the induced metric. Besides flat minimal surfaces in spheres, direct sums of surfaces in the associated family of pseudoholomorphic curves in $\mathbb{S}^5$ do satisfy this Ricci-like condition. Surfaces in both classes are exceptional …
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 consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting with any $U(n)$-invariant Kähler metric. It is known that the Kähler-Ricci flow must develop Type I singularities. We show that if the total volume does not go to zero at the singular time, then any Type I parabolic blow-up limit of the Ricci flow along the exceptional divisor is the unique $U(n)$-complete shrinking Kähler-Ricci soliton on $\mathbb C^n$ blown-up at one point. This establishes the conjecture of Feldman-Ilmanen-Knopf.
We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds $V_2\mathbb{R}^n$ and $V_{1+k_2}\mathbb{R}^n$, with $n = 1+k_2+k_3$. We use techniques from the theory of differential equations, in particular the Poincaré compactification.
We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension $n > 3$ of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead $g_{0}$ has no necks, its curvature decays and the Hopf fibers are not…
We prove that, starting at an initial metric $g(0)=e^{2u_0}(dx^2+dy^2)$ on $\mathbb{R}^2$ with bounded scalar curvature and bounded $u_0$, the Ricci flow $\partial_t g(t)=-R_{g(t)}g(t)$ converges to a flat metric on $\mathbb{R}^2$.
