Shrinking Ricci Solitons Research Articles

Integral pinched shrinking Ricci solitons

We prove that a n-dimensional, 4≤n≤6, compact gradient shrinking Ricci soliton satisfying a Ln/2-pinching condition is isometric to a quotient of the round Sn. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe–Sobolev inequality and an improved rigidity result for integral pinched Einstein metrics.

Characterizations of the compactness of Riemannian manifolds by eigenfunctions, and a partial proof of a conjecture by Hamilton

In this paper, we deal with comparison theorems for the first eigenvalue of the Schr\{o}dinger operator, and we present some criteria for the compactness of a Riemannian manifold in terms of the eigenfunctions of its Laplacian. Firstly, we establish a comparison theorem for the first Dirichlet eigenvalue $\mu _1^D (B (p,r))$ of a given Schr\{o}dinger operator. We then prove that, for the space form $ M_K^n$ with constant sectional curvature $K$, the first eigenvalue of the Laplacian operator $\lambda _1 (M_K^n)$ is greater than the limit of the corresponding first Dirichlet eigenvalue $\lambda _1^D(B_K(p,r))$. Based on these, we present a characterization of a compact gradient shrinking Ricci soliton locally being an $n$-dim space form by the first eigenfunctions of the Laplacian operator, which gives a generalization of an interesting result by Cheng \cite {4} from 2-dim to $n$-dim. This result also gives a partial proof of a conjecture by Hamilton \cite {7} that a compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. Finally, we derive a criterion of the compactness of manifolds, which gives a partial proof of another conjecture by Hamilton \cite {6} that, if a complete Riemannian 3-manifold $(M^3, g)$ satisfies the Ricci pinching condition $Rc \ge \varepsilon Rg$, where $R > 0$ and $\varepsilon $ is a positive constant, then it is compact. In fact, our result is also true for general $n$-dim manifolds.

Twisted Ricci flow on a class of Finsler metrics and its solitons

In this paper, we study a geometric flow called twisted Ricci flow on a class of Finsler manifolds. We study some solitons based on the structure of the metric. The equations of the soliton are related to Killing vector fields. This forces a shrinking Ricci soliton of a Riemannian metric to be Einstein in some cases. At last, we prove the short time existence.

Analytic and geometric properties of generic Ricci solitons

The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either $\mathds{S}^3$, $\erre\times\mathds{S}^2$ or $\erre^3$, under some very weak conditions on the vector field $X$ generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the $X$-Laplacian on every generic Ricci soliton, without any assumption on $X$.

The Soliton-Ricci Flow with variable volume forms

Abstract We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previouswork.We still call this new flow, the Soliton-Ricci flow. It corresponds to a forward Ricci type flow up to a gauge transformation. This gauge is generated by the gradient of the density of the volumes. The new Soliton-Ricci flow exist for all times. It represents the gradient flow of Perelman’s W functional with respect to a pseudo-Riemannian structure over the space of metrics and normalized positive volume forms. We obtain an expression of the Hessian of the W functional with respect to such structure. Our expression shows the elliptic nature of this operator in the orthogonal directions to the orbits obtained by the action of the group of diffeomorphism. In the case that initial data is Kähler, the Soliton-Ricci flow over a Fano manifold preserves the Kähler condition and the symplectic form. Over a Fano manifold, the space of tamed complex structures embeds naturally, via the Chern-Ricci map, into the space of metrics and normalized positive volume forms. Over such space the pseudo-Riemannian structure restricts to a Riemannian one. We perform a study of the sign of the restriction of the Hessian of the W functional over such space. This allows us to obtain a finite dimensional reduction of the stability problem for Kähler-Ricci solitons. This reduction represents the solution of this well known problem. A less precise and less geometric version of this result has been obtained recently by the author in [28].

A bound for the order of the fundamental group of a complete noncompact Ricci shrinker

For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.

Remark on a diameter bound for complete Riemannian manifolds with positive Bakry–Émery Ricci curvature

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry–Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (2009) [12] and Limoncu (2012) [8]. As an application, we shall give an upper diameter bound for compact shrinking Ricci solitons in terms of the maximum value of the scalar curvature. By using such a diameter bound, we shall provide some new sufficient conditions for four-dimensional compact shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality.

On compact Ricci solitons in Finsler geometry

Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work, it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group, and hence the first de Rham cohomology group vanishes.

Some curvature pinching results for Riemannian manifolds with density

In this note we consider versions of both Ricci and sectional curvature pinching for Riemannian manifold with density. In the Ricci curvature case the main result implies a diameter estimate that is new even for compact shrinking Ricci solitons. In the case of sectional curvature we prove a new sphere theorem.

Some curvature identities on an almost conformal gradient shrinking RICCI soliton

In this paper we have introduced the notion of almost conformai gradient shrinking Ricci soliton and established some curvature identities for almost conformai gradient shrinking Ricci soliton.

Rigidity of asymptotically conical shrinking gradient Ricci solitons

We show that if two gradient shrinking Ricci solitons are asymptotic along some end of each to the same regular cone $((0,\infty) \times \Sigma, dr^2 + r^2 g_{\Sigma} )$, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on $\mathbb{R}^n$.

Pseudo anti-commuting and Ricci soliton real hypersurfaces in complex two-plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti-commuting for real hypersurfaces in complex two-plane Grassmannians G 2 ( C m + 2 ) and prove a complete classification theorem, which gives a shrinking Ricci soliton with potential Reeb flow on Hopf real hypersurfaces and a tube over a totally real totally geodesic Q P n , m = 2 n in G 2 ( C m + 2 ) .

Stability and instability of Ricci solitons

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 (Haslhofer and Müller, arXiv:1301.3219 , 2013; Kröncke, arXiv:1312.2224 , 2013).

On Four-Dimensional Anti-self-dual Gradient Ricci Solitons

In this note we prove that any four-dimensional half conformally flat gradient steady Ricci soliton must be either Bryant's soliton or Ricci flat. We also classify four-dimensional half conformally flat gradient shrinking Ricci solitons with bounded curvature.

Remarks on compact shrinking Ricci solitons of dimension four

In this paper, we study the topological restriction of gradient shrinking Ricci solitons (M,g) of dimension 4. Let s be the scalar curvature of the metric g. Then we have:∫Msdvg=4ρvol(M), where ρ>0 is the shrinking constant and vol(M) is the volume of (M,g). We also have two kinds of topology results. (1) If we assume that:∫Ms2⩽24ρ2vol(M), then2χ(M)±3τ(M)⩾0. (2) If (M,g) is a natural oriented Kähler surface, then we have:2χ(M)+3τ(M)=ρ2vol(M)2π2. Actually, we shall show that the assumption in (1) above is equivalent to the fact that:∫Mσ2(Rc−s6g)⩾0. Here σ2(A):=σ2(g) is the 2nd symmetric function of the eigenvalues of the matrix A:=Rc−s6g.

On Moduli Spaces of Ricci Solitons

We study deformations of shrinking Ricci solitons on a compact manifold M, generalizing the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S f inside the space of all Riemannian metrics on M, we define the infinitesimal solitonic deformations and the local solitonic pre-moduli spaces. We prove the existence of a finite-dimensional submanifold of $S_{f}\times\mathcal{C}^{\infty}(M)$ , which contains the pre-moduli space of solitons around a fixed shrinking Ricci soliton as an analytic subset. We define solitonic rigidity and give criteria which imply it.

Perelman's entropy functional at Type I singularities of the Ricci flow

Abstract We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's 𝒲-functional. First, we give an alternative proof of the result obtained by Naber (2010) and Enders–Müller–Topping (2011) that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result relates a limit 𝒲-density at a Type I singular point to the entropy of the limit gradient shrinking soliton obtained by blowing-up at this point. In particular, we show that no entropy is lost at infinity during the blow-up process.

On Bach-flat gradient shrinking Ricci solitons

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton $R^4$ or the round cylinder $S^3\times R$. More generally, for n>4, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton $R^n$ or the product $N^{n-1}\times R$, where $N^{n-1}$ is Einstein.

Cohomogeneity one shrinking Ricci solitons: an analytic and numerical study

We use analytical and numerical methods to investigate the equations for cohomogeneity one shrinking gradient Ricci solitons. We show the existence of a winding number for this system around the subvariety of phase space corresponding to Einstein solutions and obtain some estimates for it. We prove a non-existence result for certain orbit types, analogous to that of Bohm in the Einstein case. We also carry out numerical investigations for selected orbit types.

LOWER BOUND ESTIMATES OF THE FIRST EIGENVALUE FOR THE f-LAPLACIAN AND THEIR APPLICATIONS

In this paper, we first establish some new lower bounds for the first eigenvalue of the f -Laplacian on closed Riemannian manifolds with the N -Bakry–Emery Ricci curvature satisfying RicNf ≥ K (K ∈ R), which not only improve the classical results of lower bounds for the first eigenvalue of the Laplacian, but also are suitable for both smooth metric spaces and quasi-Einstein manifolds. Then we obtain the corresponding lower bounds for closed manifolds with Ricf ≥ K (K ∈ R), which include gradient Ricci solitons. As an application, we derive a universal lower bound for the diameter of compact non-trivial shrinking Ricci solitons, which improves the results ofAndrews–Ni [1] and Futaki–Sano [9].