Gradient Shrinking Ricci Solitons Research Articles

Geometry of shrinking Ricci solitons

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\text{Rm}$ of $M$ satisfies the estimate $|\text{Rm}|\leqslant cS$ for some constant $c$. Moreover, the curvature operator $\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where $r$ is the distance function to a fixed point in $M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

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 $\mathbb{R}^n$.

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.

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.

On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten–Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first author and Sano (Asian J Math, to appear), and by Andrews and Ni (Comm Partial Differential Equ, to appear). Moreover, we extend the diameter estimate to compact self-similar shrinkers of mean curvature flow.

Volume estimate about shrinkers

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.

Nonexistence of nontrivial quasi-Einstein metrics

Let (Mn, g, e–f dvolg) be a smooth metric measure space of dimension n. In this note, we first prove a nonexistence result for Mn with the Bakry-Émery Ricci tensor is bounded from below. Then we show that f $\in$ L∞ (Mn, e–f dvol) and |∇f| $\in$ L∞ (Mn, e–f dvol) are equivalent for complete gradient shrinking Ricci solitons. Furthermore, we prove that there is no non-Einstein shrinking soliton when the normalized function $\tilde f$ is non-positive.

Complete gradient shrinking Ricci solitons with pinched curvature

We prove that any n-dimensional complete gradient shrinking Ricci soliton with pinched Weyl curvature is a finite quotient of \({\mathbb{R}^{n}, \mathbb{R}\times \mathbb{S}^{n-1}}\) or \({\mathbb{S}^{n}}\) . In particular, we do not need to assume the metric to be locally conformally flat.

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

Following Li and Yau (Acta Math 156:153–201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional $${\mathcal{J}}$$ associated to a smooth function $${\phi}$$ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.

A necessary and sufficient condition for Ricci shrinkers to have positive AVR

In this short paper we observe that a recent result of C.-W. Chen meshes well with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L. Ni, and S.-J. Zhang. We give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci solitons to have positive asymptotic volume ratio.

On Gradient Ricci Solitons

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work in M. Fernandez-Lopez and E. Garcia-Rio, Rigidity of shrinking Ricci solitons in Math. Z. (2011) classify complete gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking Kahler and gradient steady Ricci solitons. Consequences to the structure of shrinking and steady solitons at infinity are also discussed.

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelmanʼs previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.

Maximum principles and gradient Ricci solitons

It is shown that the Omori–Yau maximum principle holds true on complete gradient shrinking Ricci solitons both for the Laplacian and the f-Laplacian. As an application, curvature estimates and rigidity results for shrinking Ricci solitons are obtained. Furthermore, applications of maximum principles are also given in the steady and expanding situations.

The Fundamental Groups of m-Quasi-Einstein Manifolds

In Ricci flow theory, the topology of Ricci soliton is important. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of gradient shrinking Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of m-quasi-Einstein with a positive constant multiple.

ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.

The curvature of gradient Ricci solitons

Our goal in this paper is to obtain further information about the curvature of gradient shrinking Ricci solitons. This is important for a better understanding and ultimately for the classification of these manifolds. The classification of gradient shrinkers is known in dimensions 2 and 3, and assuming locally conformally flatness, in all dimensions n ≥ 4 (see [14, 13, 6, 15, 20, 12, 2]). Many of the techniques used in these works required some control of the Ricci curvature. For example, in [15] gradient shrinking Ricci solitons which are locally conformally flat were classified assuming an integral condition on the Ricci tensor. This condition and other integral estimates of the curvature were later proved in [12]. Without making the strong assumption of being conformally flat, it is natural to ask whether similar estimates are true for the Riemann curvature tensor. In this paper we are able to prove pointwise estimates on the Riemann curvature, assuming in addition that the Ricci curvature is bounded. We will show that any gradient shrinking Ricci soliton with bounded Ricci curvature has Riemann curvature tensor growing at most polynomially in the distance function. We note that by Shi’s local derivative estimates we can then obtain growth estimates on all derivatives of the curvature. This, in particular, proves weighted L estimates for the Riemann curvature tensor and its covariant derivatives. We point out that for self shrinkers of the mean curvature flow Colding and Minicozzi [8] were able to prove weighted L estimates for the second fundamental form, assuming the mean curvature is positive. These estimates were instrumental in the classification of stable shrinkers. Our estimates can be viewed as parallel to theirs, however the classification of gradient Ricci solitons is still a major open question in the field. A gradient shrinking Ricci soliton is a Riemannian manifold (M, g) for which there exists a potential function f such that