TITLE: On Complete Non-compact Ricci-flat Cohomogeneity One Manifolds

We present an alternative proof of the existence theorem of Böhm using ideas from the study of gradient Ricci solitons on the multiple warped product cohomogeneity one manifolds by Dancer and Wang. We conclude that the complete Ricci-flat metric converges to a Ricci-flat cone. Also, starting from a 4n-dimensional HPn base space, we construct numerical Ricci-flat metrics of cohomogeneity one in (4n + 3) dimensions whose level surfaces are CP 2n+1. We show the local Ricci-flat solution is unique (up to homothety). The numerical results suggest that they all converge to Ricci-flat Ziller cone metrics even if n = 2. i Acknowledgement This thesis is the product of two years of enjoyable and productive collaboration with my advisor, McKenzie Wang. I thank him for his confidence in me, for generously sharing his talents, energy, and good advice, for so many hours of his time, and most of all for introducing me to differential geometry. This thesis is a testimony to his good will and hard work. I thank Ian Hambleton and Andrew Nicas for generously sharing their expertise and for encouragement throughout my graduate studies. I also want to thank my undergraduate teacher Weixiao Shen and Jiansong Deng