Related papers: A new Li-Yau-Hamilton estimate for Kahler-Ricci fl…

In this note we prove a new \epsilon-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into…

Let $(M,J_0)$ be a Fano manifold which admits a K\"ahler-Ricci soliton, we analyze the behavior of the K\"ahler-Ricci flow near this soliton as we deform the complex structure $J_0$. First, we will establish an inequality of Lojasiewicz's…

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $\kappa$ solutions to the Ricci flow. As an application, using the $W$ entropy associated with the heat…

We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano K\"ahler-Ricci flow to general finite time solutions of the K\"ahler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates…

In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate…

We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type…

This paper provides a first contribution to port-Hamiltonian modeling of district heating networks. By introducing a model hierarchy of flow equations on the network, this work aims at a thermodynamically consistent port-Hamiltonian…

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit…

In this paper, the author has considered the hyperbolic Kahler-Ricci flow introduced by Kong and Liu [11], that is, the hyperbolic version of the famous Kahler-Ricci flow. The author has explained the derivation of the equation and…

$\mathscr{W}$-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static…

In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type…

In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of K\"ahler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local…

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a K\"ahler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an…

We first prove a uniform integral Laplace comparison result for the K\"ahler Ricci flow on Fano manifolds which depends only on the initial metric. As an application, using Cheeger-Colding theory and previous results by some of the authors,…

In this paper, we extend the theory of Ricci flows satisfying a Type-I scalar curvature condition at a finite-time singularity. In [Bam16], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci…

The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau, the…

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_t u=Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds…

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone…

The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold $M$ with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to…

In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a K\"ahler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is…