Davies type estimate and the heat kernel bound under the Ricci flow

We prove a Davies type double integral estimate for the heat kernel $H(y,t;x,l)$ under the Ricci flow. As a result, we give an affirmative answer to a question proposed by Chow etc.. Moreover, we apply the Davies type estimate to provide a new proof of the Gaussian upper and lower bounds of $H(y,t;x,l)$ which were first shown by Chau-Tam-Yu.Comment: conditions refined, some errors correcte

The second variation of the Ricci expander entropy

We compute the second variation of the Ricci expander entropy and briefly discuss the linear stability of compact negative Einstein manifolds

On rigidity of gradient K\"ahler-Ricci solitons with harmonic Bochner tensor

In this paper, we prove that complete gradient steady K\"ahler-Ricci solitons with harmonic Bochner tensor are necessarily K\"ahler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) K\"ahler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times \mathbb{C}^{n-k}$, where $N$ is a K\"ahler-Einstein manifold with positive (or negative) scalar curvature.Comment: minor errors correcte

Ricci solitons on Sasakian manifolds

We show that a Sasakian metric which also satisfies the gradient Ricci soliton equation is necessarily Einstein.Comment: 4 page

Light pseudoscalar meson and doubly charmed baryon scattering lengths with heavy diquark-antiquark symmetry

We adopt the heavy baryon chiral perturbation theory (HBChPT) to calculate the scattering lengths of $\phi B_{cc}^{(*)}$ up to $\mathcal{O}(p^3)$, where $\phi$ is the pseudoscalar mesons. The recoil effect and the mass splitting between the spin-$1\over 2$ and spin-$3\over 2$ doubly charmed baryons are included. In order to give the numerical results, we construct the chiral Lagrangians with heavy diquark-antiquark (HDA) symmetry in a formally covariant approach. Then, we relate the low energy constants (LECs) of the doubly charmed baryons to those of $D^{(*)}$ mesons. The LECs for the $\phi D^{(*)}$ scattering are estimated in two scenarios, fitting lattice QCD results and using the resonance saturation model. The chiral convergence of the first scenario is not good enough due to the the large strange quark mass and the presence of the possible bound states, virtual states and resonance. The final results for two scenarios are consistent with each other. The interaction for the $[\pi\Xi^{(*)}_{cc}]^{(1/2)}$, $[K\Xi^{(*)}_{cc}]^{(0)}$, $[K\Omega^{(*)}_{cc}]^{(1/2)}$, $[\eta\Xi^{(*)}_{cc}]^{(1/2)}$, $[\eta\Omega^{(*)}_{cc}]^{(0)}$ and $[\bar{K}\Xi^{(*)}_{cc}]^{(0)}$ channels are attractive. The most attractive channel $[\bar{K}\Xi^{(*)}_{cc}]^{(0)}$ may help to form the partner states of the $D_{s0}^*(2317)$ ($D_{s1}(2460)$) in the doubly heavy sector.Comment: 12 pages, 2 figur

On second variation of Perelman's Ricci shrinker entropy

In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman's $\nu$-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly stable shrinkers in terms of the least eigenvalue and its multiplicity of certain Lichnerowicz type operator associated to the second variation.Comment: 13 pages; final version; to appear in Math. An

New Volume Comparison results and Applications to degeneration of Riemannian metrics

We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-\'Emery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-\'Emery Ricci curvature condition, we initiate an approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the {\it original manifold}. These results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead. Applications on convergence and degeneration of Riemannian metrics under this curvature condition are given. To this effect, in particular for the Bakry-\'Emery Ricci curvature condition, the gradient of the potential function is allowed to have singularity of order close to $1$ while the traditional method of weighted manifolds allows bounded gradient. This approach enables us to extend some of the results in the papers \cite{Co}, \cite{ChCo2}, \cite{zZh}, \cite{TZ} and \cite{WZ}. The condition also covers general Ricci solitons instead of just gradient Ricci solitons.Comment: 45 page

Aronson-B\'enilan estimates for the fast diffusion equation under the Ricci flow

We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-B\'enilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting

Aronson-B\'enilan estimates for the porous medium equation under the Ricci flow

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-B\'enilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the Ricci flow.Comment: Minor changes to the abstract and remark 1.

Bounds on harmonic radius and limits of manifolds with bounded Bakry-\'Emery Ricci curvature

Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $C^{\alpha} W^{1, q}$ harmonic radius for manifolds with bounded Bakry-\'Emery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $C^\alpha W^{1,q}$, where $q>2n$ and $n$ is the dimension of the manifolds. This is almost 1 order lower than that in the classical $C^{1,\alpha} W^{2, p}$ harmonic coordinates under bounded Ricci curvature condition [And]. The loss of regularity induces some difference in the method of proof, which can also be used to address the detail of $W^{2, p}$ convergence in the classical case. Based on this lower bound and the techniques in [ChNa2] and [WZ], we extend Cheeger-Naber's Codimension 4 Theorem in [ChNa2] to the case where the manifolds have bounded Bakry-\'Emery Ricci curvature when the gradient of the potential is bounded. This result covers Ricci solitons when the gradient of the potential is bounded. During the proof, we will use a Green's function argument and adopt a linear algebra argument in [Bam]. A new ingradient is to show that the diagonal entries of the matrices in the Transformation Theorem are bounded away from 0. Together these seem to simplify the proof of the Codimension 4 Theorem, even in the case where Ricci curvature is bounded.Comment: 35pages; modified after suggestions by a number of peopl