The Sasaki–Ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature

We show that Perelman’s ${\mathcal{W}}$ functional on Kähler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kähler–Ricci flow (the first Chern class is positive) can be generalized to Sasaki–Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectional curvature is preserved along Sasaki–Ricci flow, using Bando and Mok’s methods and results in Kähler–Ricci flow. In particular, we show that the Sasaki–Ricci flow converges to a Sasaki–Ricci soliton when the initial metric has nonnegative transverse bisectional curvature.     

Positive isotropic curvature and self-duality in dimension 4

We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: the half-PIC condition. It is a slight weakening of the positive isotropic curvature (PIC) condition introduced by M. Micallef and J. Moore. We observe that the half-PIC condition is preserved by the Ricci flow and satisfies a maximality property among all Ricci flow invariant positivity conditions on the curvature of oriented 4-manifolds. We also study some geometric and topological aspects of half-PIC manifolds.     

<Emphasis Type="Italic">W</Emphasis>-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman’s seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K,m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a (K,m)-super Ricci flow, where K ∈ R and m ∈ [n,∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result improves an important result due to Lott and Villani (2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.     

On the extension of the Harmonic Ricci flow

In this short paper, we prove that if the Ricci curvature is uniformly bounded under the Harmonic Ricci flow on complete manifolds for time ${t\in [0,T)}$ , then the Harmonic Ricci flow can be extended past the time T.     

Remarks on Kähler Ricci Flow

We show the convergence of Kähler Ricci flow directly if the α-invariant of the canonical class is greater than \(\frac{n}{n+1}\). Applying these convergence theorems, we can give a Kähler Ricci flow proof of Calabi conjecture on such Fano manifolds. In particular, the existence of KE metrics on a lot of Fano surfaces can be proved by flow method. Note that this geometric conclusion (based on the same assumption) was established earlier via elliptic method by Tian (Invent. Math. 89(2):225–246, 1987; Invent. Math. 101(1):101–172, 1990; Invent. Math. 130:1–39, 1997). However, a new proof based on Kähler Ricci flow should be still interesting in its own right.     

Eigenvalues of $$(-\triangle + \frac{R}{2})$$ on manifolds with nonnegative curvature operator

In this paper, we show that the eigenvalues of are nondecreasing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat.     

Evolution Equations of Curvature Tensors Along the Hyperbolic Geometric Flow

The author considers the hyperbolic geometric flow δ2/δt2 g(t) =-2Ricg(t) introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms of evolution equations for Levi-Civita connection and curvature tensors under the hyperbolic geometric flow. In addition, similarly to the Ricci flow case, it is shown that any solution to the hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.     

Some characterizations of space-forms

Integral conditions on the traceless Ricci tensor are used to characterize Euclidean and hyperbolic spaces among complete, locally conformally flat manifolds of constant scalar curvature. The main tools are vanishing-type results for -solutions of a large class of differential inequalities. Further applications of the technique are also given.

     

Normalized Ricci flows and conformally compact Einstein metrics

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in Graham and Lee (Adv Math 87:186–255, 1991), Lee (Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds, 2006), and Biquard (Surveys in Differential Geometry: Essays on Einstein Manifolds, 1999) of conformally compact Einstein metrics with conformal infinities which are perturbations of that of given non-degenerate conformally compact Einstein metrics.     

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

Following Li and Yau (Acta Math 156:153?C201 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.     

Volume Comparison of Conformally Compact Manifolds with Scalar Curvature <Emphasis Type="Italic">R</Emphasis> ≥ −<Emphasis Type="Italic">n</Emphasis> (<Emphasis Type="Italic">n</Emphasis> − 1)

In this paper, we use the normalized Ricci–DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature R ≥ ?n (n ? 1) and also the rigidity result when certain relative volume is zero.     

Ricci Flow on a Class of Noncompact Warped Product Manifolds

We consider the Ricci flow on noncompact \(n+1\)-dimensional manifolds M with symmetries, corresponding to warped product manifolds \(\mathbb {R}\times T^n\) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate \(|{{\mathrm{Rm}}}|(p,t) \le C/t\) for some \(C > 0\) and all \(p \in M, t \in (1,\infty )\) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as \(t \rightarrow \infty \)) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.     

Evolution of the first eigenvalue of the clamped plate on manifold along the Ricci flow

ABSTRACT

In this article, we study the evolution, monotonicity for the first eigenvalue of the clamped plate on closed Riemannian manifold along the Ricci flow. We prove that the first nonzero eigenvalue is nondecreasing under the Ricci flow under certain geometric conditions and find some applications in 2-dimensional manifolds.     

On the Perelman’s reduced entropy and Ricci flat manifolds with maximal volume growth

In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman’s reduced volume of Ricci flow. We show that if $(M^n,g)$ is an noncompact complete Ricci flat manifold with maximal volume growth satisfying $|Rm|(x)\rightarrow 0$ as $d(x)=d_g(x,p)\rightarrow \infty $ , then $M^n$ has the quadratic curvature decay. Some applications to this result are also presented.     

Gluing Eguchi‐Hanson Metrics and a Question of Page

In 1978, Gibbons‐Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with 16 orbifold points and resolves the orbifold singularities by gluing in 16 small Eguchi‐Hanson manifolds that all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun‐Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi‐Hanson manifolds. This is a subtle question: if successful, this construction would produce Einstein metrics that are neither Kähler nor self‐dual. In this paper, we focus on a configuration of maximal symmetry involving eight small Eguchi‐Hanson manifolds of each orientation that are arranged according to a chessboard pattern. By analyzing the interactions between Eguchi‐Hanson manifolds with opposite orientation, we identify a nonvanishing obstruction to the gluing problem, thereby destroying any hope of producing a metric of zero Ricci curvature in this way. Using this obstruction, we are able to understand the dynamics of such metrics under Ricci flow as long as the Eguchi‐Hanson manifolds remain small. In particular, for the configuration described above, we obtain an ancient solution to the Ricci flow with the property that the maximum of the Riemann curvature tensor blows up at a rate of , while the maximum of the Ricci curvature converges to 0.© 2016 Wiley Periodicals, Inc.     

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.     

A sub-Riemannian curvature-dimension inequality,volume doubling property and the Poincaré inequality

Let $\mathbb M $ be a smooth connected manifold endowed with a smooth measure $\mu $ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$ , and which is symmetric with respect to $\mu $ . We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold:

• The volume doubling property;
• The Poincaré inequality;
• The parabolic Harnack inequality.

The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.     

Harnack Estimates for Ricci Flow on a Warped Product

In this paper, we study the Ricci flow on a closed manifold equipped with a warped product metric \((N\times F,g_{N}+f^2 g_{F})\) in which \((F,g_{F})\) is Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of Perelman’s differential Harnack inequality (The entropy formula for the Ricci flow and its geometric applications, 2002).     

Harnack inequality for degenerate and singular operators of <Emphasis Type="Italic">p</Emphasis>-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds. The Krylov–Safonov type Harnack inequality for the p-Laplacian operators with \(1<p<\infty \) is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear p-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.