UC San Diego

In this dissertation we study two problems related to Ricci flow on complete noncompact manifolds. In Chapter 1 we report joint work with Ben Chow and Peng Lu on volume growth, lower bounds on scalar curvature, and upper bounds on Ricci curvature for complete gradient Ricci solitons. Most of our results are obtained without any assumption on the soliton metrics. We wish they could be useful for further study on complete gradient Ricci solitons under general assumptions. In Chapter 2 we obtain several results on complete Kähler manifolds with nonnegative curvature. The uniformization problem of such manifolds has been an important topic in complex geometry. We construct new examples of such metrics and discuss their connection with Kähler-Ricci flow. This is partly joint work with Fangyang Zheng. We also explore the interaction between function theory and metric geometry on general Kähler manifolds with nonnegative bisectional curvature. This part is selected from my ongoing project and complete results will appear elsewhere