Abstract:
In this paper we prove a new matrix Li-Yau-Hamilton estimate for K\"ahler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate and its linear version proved earlier in \cite{CN}. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian, a manifold version of Stoll's theorem on the characterization of `algebraic divisors', and a localized monotonicity formula for analytic subvarieties. Motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of Perelman, we prove a sharp lower bound heat kernel estimate for the time-dependent heat equation, which is, in a certain sense, dual to Perelman's monotonicity of the `reduced volume'. As an application of this new monotonicity formula, we show that the blow-down limit of a certain type III immortal solution is a gradient expanding soliton. In the last section we also illustrate the connection between the new Li-Yau-Hamilton estimate and the earlier Hessian comparison theorem on the `reduced distance', proved in \cite{FIN}.

Abstract:
In this paper, we prove that any non-flat ancient solution to K\"ahler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional curvature must be compact. Both results generalize the corresponding earlier results of Perelman in \cite{P1} and \cite{P2}. The results can be applied to study the geometry and function theory of complete K\"ahler manifolds with nonnegative bisectional curvature via K\"ahler-Ricci flow. It also implies a compactness theorem on ancient solutions to K\"ahler-Ricci flow.

Abstract:
We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.

Abstract:
We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation with respect to evolving Riemannian metrics via a space-time consideration. Some new monotonicity formulae are derived. As applications of the new local monotonicity formulae, some local regularity theorems concerning Ricci flow are proved.

Abstract:
By adapting methods of \cite{AC} we prove a sharp estimate on the expansion modulus of the gradient of the log of the parabolic kernel to the Sch\"ordinger operator with convex potential, which improves an earlier work of Brascamp-Lieb. We also include alternate proofs to the improved log-concavity estimate, and to the fundamental gap theorem of Andrews-Clutterbuck via the elliptic maximum principle. Some applications of the estimates are also obtained, including a sharp lower bound on the first eigenvalue.

Abstract:
We show a connection between the linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow and the monotonicity formula for the positive currents. As an application of the linear trace Li-Yau-Hamilton stated in this paper and the one proved by Chow-Hamiltonm, we give another proof on the classification of the limit solutions of the Kaehler-Ricci (Ricci) flow.

Abstract:
In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. The example is the first of this type. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow.

Abstract:
We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman's recent results on volume non-collapsing.

Abstract:
In this paper, we derive a new monotonicity formula for the plurisuhbarmonic functions on complete K\"ahler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of holomorphic functions (sections) with polynomial growth, which in particular, partially solve a conjecture of Yau.

Abstract:
By solving the Cauchy problem for the Hodge-Laplace heat equation for $d$-closed, positive $(1, 1)$-forms, we prove an optimal gap theorem for K\"ahler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius $r$ centered at any fixed point $o$ is a function of $o(r^{-2})$. Furthermore via a relative monotonicity estimate we obtain a stronger statement, namely a `positive mass' type result, asserting that if $(M, g)$ is not flat, then $\liminf_{r\to \infty} \frac{r^2}{V_o(r)}\int_{B_o(r)}\mathcal{S}(y)\, d\mu(y)>0$ for any $o\in M$.