Rigidity Of Gradient Ricci Solitons With Harmonic Weyl Curvature And Related Problems

The concept of Ricci solitons was introduced by R.Hamilton in the mid1980's.They are also special solutions to Hamilton's Ricci flow and play important roles in the singularity study of the Ricci flow.Ricci solitons are also natural generalizations of Einstein metrics.In this thesis,we mainly study the geometric structure of gradient Ricci solitons and related topics under some curvature conditions,including complete gradient Ricci solitons with harmonic Weyl curvature,vacuum static spaces and CPE metrics with harmonic curvature,as well as the Myers' type theorem for integral Bakry-(?)mery Ricci tensor bounds.More precisely,in the first chapter,we will review the research background and recent progress,and show our main work of rigidity of complete gradient steady Ricci solitons and related problems.In the second chapter,our main aim is to investigate the rigidity of complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor.More precisely,we prove that an n-dimensional(n ? 5)complete noncompact gradient steady Ricci soliton with harmonic Weyl tensor is either Ricci flat or isometric to the Bryant soliton up to scaling.We also derive a classification result for complete noncompact gradient expanding Ricci solitons with harmonic Weyl tensor.Meanwhile,for n ? 5,we provide a local structure theorem for n-dimensional connected(not necessarily complete)gradient Ricci solitons with harmonic Weyl curvature,thus extending the work of J.Kim [77] for n = 4.In the third chapter,we classify n-dimensional(n ? 5)vacuum static spaces with harmonic curvature,thus extending the 4-dimensional work by J.Kim-J.Shin [78].As a consequence,we provide new counterexamples to the FischerMarsden conjecture on compact vacuum static spaces.In the fourth chapter,our main aim is to investigate Besse Conjecture stated that any compact CPE space is necessarily Einstein,but it is not true.We give the classification complete CPE metrics with harmonic curvature,of which is equivalent to be D-flat and of constant scalar curvature,and then provide counterexamples to the Besse conjecture.Meanwhile,critical spaces of harmonic curvature can be classified.Finally,it turns out that the Besse conjecture holds if and only if the trcae-free Ricci mean value vanishes of a compact CPE metric.In the last chapter,we study the diameter estimate of smooth metric measure space.We first correct comparison Theorems [124].Then,we apply the corrected comparison results to get a new Myers' type theorem for integral Bakry-(?)mery Ricci tensor bounds,which can be viewed as the extension of the works of S.Myers and E.Aubry.