On Complete Non-compact Ricci-flat Cohomogeneity One Manifolds

Abstract

<p>We present an alternative proof of the existence theorem of B\"ohm 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 $\mathbb{H}P^{n}$ base space, we construct numerical Ricci-flat metrics of cohomogeneity one in ($4n+3$) dimensions whose level surfaces are $\mathbb{C}P^{2n+1}$. 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 $n=2$.</p>