Nonpostively curved Hermitian metric on product manifolds

In this article, we classify all the Hermitian metrics on a complex product manifold with nonpositive holomorphic bisectional curvature. It is a generalization of a result by Zheng.Comment: 4 page

Time-dependent Sobolev inequality along the Ricci flow

In this article we get a time-dependent Sobolev inequality along the Ricci flow which generalizes the earlier results of Zhang, Ye, Hsu. As an application of the time-dependent Sobolev inequality, we also get a growth of the ratio of bob-collapsing along the Ricci flow.Comment: Proof of Theorem 2.2 is added, a serious mistake in the proof of Theorem 2.3 is correcte

Curvature of almost Hermitian manifolds and applications

In this paper, by introducing a notion of local quasi holomorphic frame, we obtain a curvature formula for almost Hermitian manifolds which is similar to that of Hermitian manifolds. Moreover, as applications of the curvature formula, we extend a result of H.S. Wu and a result of F. Zheng to almost Hermitian manifolds.Comment: 15 page

Curvature identities on almost Hermitian manifolds and applications

In this paper, we systematically compute the Bianchi identities for the canonical connection on an almost Hermitian manifold. Moreover, we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection. As applications of the curvature identities, we obtain some results about the integrability of quasi Kaehler manifolds and some properties of nearly Kaehler manifolds.Comment: 15 pages, a mistake in the published version in Theorem 3.5 was fixe

Hessian comparison and eigenvalue estimate of almost Hermitian manifolds

In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain a sharp eigenvalue estimate on quasi Kaehler manifolds and a sharp Hessian comparison on nearly Kaehler manifolds.Comment: 20 page

Negativity of Perelman's Li-Yau-Hamilton type expression

Chau-Tam-Yu has proved the non-positivity of Perelman's new Li-Yau-Hamilton type expression on noncompact manifolds. In this article, we further prove that $v$ is negative if the Ricci flow is not end up with an Euclidean space.Comment: 4 page

A Note on Wu-Zheng's Splitting Conjecture

Cao's splitting theorem says that for any complete K\"ahler-Ricci flow $(M,g(t))$ with $t\in [0,T)$, $M$ simply connected and nonnegative bounded holomorphic bisectional curvature, $(M,g(t))$ is holomorphically isometric to \C^k\times (N,h(t)) where $(N,h(t))$ is a Kahler-Ricci flow with positive Ricci curvature for $t>0$. In this article, we show that $k=n-r$ where $r$ is the Ricci rank of the initial metric. As a corollary, we also confirm a splitting conjecture of Wu-Zheng when curvature is assumed to be bounded

A Liouville property of holomorphic maps

In this article, we prove a Liouville property of holomorphic maps from a complete Kahler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kahler manifold with a certain assumption on the sectional curvature.Comment: 4 page

A geometric proof of the classification of complex vector cross product

In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.Comment: 3 page

Heat kernel recurrence on space forms and applications

In this paper, we first give a direct proof for two recurrence relations of the heat kernels for hyperbolic spaces in \cite{DM}. Then, by similar computation, we give two similar recurrence relations of the heat kernels for spheres. Finally, as an application, we compute the diagonal of heat kernels for odd dimensional hyperbolic spaces and the heat trace asymptotic expansions for odd dimensional spheres.Comment: 17pages; All comments are welcom