Recent developments of the DDVV Conjecture

In this paper, we give a survey of the recent develpoments of the DDVV conjecture

On the Geometry of Classifying Spaces and Horizontal Slices

In this paper, we study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let $U$ be a neighborhood of the moduli space. Then we know the universal covering space $V$ of $U$ is a smooth manifold. Suppose $D$ is the classifying space of a polarized Calabi-Yau manifold with the automorphism group $G$. Let $D_1$ be the symmetric space associated with $G$. Then we proved that the map from $V$ to $D_1$ induced by the period map is a pluriharmonic map. We also give a Kahler metric on $V$, which is called the Hodge metric. We proved that the Ricci curvature of the Hodge metric is negative away from zero. We also proved the non-existence of the K\"ahler metric on the classifying space of a Calabi-Yau threefold which is invariant under a cocompact lattice of $G$

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds

In this paper, we represent the Hodge metric in terms of the Weil-Petersson metric and its Ricci curvature on the moduli spaces of polarized Calabi-Yau threefolds

On the ground state of quantum layers

We provide some new results of the ground state of quantum layers

Remarks on the Bottcher-Wenzel Inequality

In 2005, B\"ottcher and Wenzel raised the conjecture that if $X,Y$ are real square matrices, then $||XY-YX||^2\leq 2||X||^2||Y||^2$, where $||\cdot||$ is the Frobenius norm. Various proofs of this conjecture were found in the last few years by several authors. We here give another proof. This proof is highly conceptual and requires minimal computation. We also briefly discuss related inequalities, in particular, the classical Chern-do Camo-Kobayashi inequality.Comment: Journal reference adde

On the Lower Order Terms of the Asymptotic Expansion of Zelditch

In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch. We also proved that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight $k$.Comment: 34 page

A Note on Special Kahler Manifolds

We proved a conjecture of D. Freed that there are no non-trivial complete special Kaehler manifolds.Comment: 3 page

Proof of the normal scalar curvature conjecture

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture

Gradient estimates of the Yukawa coupling

The paper is related to the classification of special manifolds and projective special manifolds. One of the result of this paper is that, if the Weil-Petersson metric on a projective special manifold is complete, then the Hodge metrc and the Weil-Petersson metrc are equivalent

K Energy and K stability on Hypersurfaces

In this paper, we study the limiting properties of the $K$ energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. K\"ahler-Einstein metrics and the generalized Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of hypersurfaces. In particular, we allow the center fiber of a special degeneration (a degeneration by a one-parameter group) to have multiplicity great than 1