In this paper we construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of $S^2 \times S^3$. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A Kähler–Einstein metric on the Del Pezzo orbifold is then lifted to an Einstein metric using the Kobayashi–Boyer–Galicki method.

In this paper we prove a new matrix Li-Yau-Hamilton (LYH) estimate for Kähler-Ricci flow on manifolds with nonnegative bisectional curvature. The form of this new LYH estimate is obtained by the interpolation consideration originated by Chow. 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 corresponding estimate for linear heat equation. 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, which sharpens the Bishop volume comparison theorem.

Motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of Perelman, we prove a sharp lower bound of the fundamental solution to the forward conjugate heat equation, which 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 of long-time solution is a gradient expanding soliton, generalizing an earlier result of Cao. We also illustrate the connection between the new LYH estimate and the Hessian comparison theorem of M. Feldman, T. Ilmanen, and L. Ni, on the forward reduced distance. Localized monotonicity formulae on entropy and forward reducedvolume are also derived.