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**1 - 7**of**7**### Einstein metrics from symmetry and bundle constructions

- in Surveys in Differential Geometry VI: Essays on Einstein Manifolds, (A Supplement to the Journal of Differential Geometry
, 1999

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### Structure of homogeneous Ricci solitons and the Alekseevskii conjecture, arXiv: 1212.6511v1

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### A Hamiltonian approach to the cohomogeneity one Ricci soliton equations, arXiv:1407.2551(math.DG

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### Contents

"... Abstract. We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space Hq,n ⊂ Λ2g ∗ ⊗ g of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic so ..."

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Abstract. We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space Hq,n ⊂ Λ2g ∗ ⊗ g of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of the identity plus a derivation) are precisely the fixed points of the system, and that a homogeneous Ricci soliton is isometric to an algebraic soliton if and only if the corresponding bracket flow solution is not chaotic, in the sense that its ω-limit set consists of a single point. We also geometrically characterize algebraic solitons among homogeneous Ricci solitons as those for which the Ricci flow solution is simultaneously

### 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 ..."

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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.