Colding and Minicozzi have shown that if an embedded minimal disk in $B_R\subset\Real^3$ has large curvature then in a smaller ball, on a scale still proportional to $R$, the disk looks roughly like a piece of a helicoid. In this talk, we will see that near points whose curvature is relatively large the description can be made more precise. That is, in a neighborhood of such a point (on a scale $s$ proportional to the inverse of the curvature of the point) the surface is bi-Lipschitz to a piece of a helicoid. Moreover, the Lipschitz constant goes to 1 as $Rs$ goes to $\infty$ . This follows from Meeks and Rosenberg's result on the uniqueness of the helicoid of which, time permitting, we will discuss a new proof. Joint work with C. Breiner.

In this talk I will discuss some general conditions such that the Poisson equation can be solved on a complete manifold. Existence of harmonic maps between complete manifolds and existence of Hermitian-Einstein metrics on holomorphic vector bundles over complete manifolds will be mentioned as applications. This is joint work with Natasa Sesum.

As the complex version of Ricci flow, K\"ahler-Ricci flow enjoys the special feature, i.e., cohomology information for the evolving K\"ahler metric. The flow can thus be reduced to scalar level as first used by H. D. Cao in the alternative proof of Calabi's Conjecture. People have mostly been focusing on the situation when the K\"ahler class is fixed. As first considered by H. Tsuji, by allowing the class to evolve, the flow can be applied in the study of degenerate class, for example, class on the boundary of K\"ahler cone. We discuss some results in this drection. This is the geometric analysis aspect of Tian's program, which aims at applying K\"ahler-Ricci flow in the study of algebraic geometry objects with great interests.

We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several
rigidity results. We also give a splitting theorem for some Kahler quasi-Einstein metrics.