RIGIDITY ON EINSTEIN MANIFOLDS AND SHRINKING RICCI SOLITONS IN HIGH DIMENSIONS

摘要

This thesis consists of three parts. Each part solves a geometric problem in geometricanalysis using differential equations.The first part gives a rigidity result to high dimensional positive Einstein manifolds,by controlling the upper bound of the integration of Weyl tensor.Part of the idea of the second part came from the new weighted Yamabe invariantfrom [4]. According to the definition, we can show a rigidity theorem to highdimensionalcompact shrinking Ricci solitons.The third part is an analytical result to 4-dimensional Ricci solitons. By theWeitzenbock for Ricci solitons introduced in [5], we proved an integral versionof the Weitzenbock formula.