Curvature Estimates And Applications For Steady And Expanding Ricci Solitons

Abstract

The interplay between curvature, geometry and topology of a manifold has long been a main theme in Riemannian geometry. In this thesis, we attempt to reveal the geometry of Ricci soliton from the curvature point of view. Ricci soliton is of great significance since it is a self similar solution to the Ricci flow and appears in the singularity analysis of the flow. It is also a natural generalization of the Einstein manifold. Firstly, we study the curvature decay of noncompact steady gradient Ricci soliton. One conjectural behavior of steady soliton is that if the curvature ever decays, then the decay rate is either linear or exponential. We shall focus on the steady soliton whose decay rate is faster than the former case. Under some conditions on the potential function, we refine an estimate by Munteanu-Sung-Wang and establish a sharp exponential curvature decay for steady gradient soliton with curvature decay rate faster than linear rate. In dimension four, we show that the curvature tensor is bounded from above by the scalar curvature if the scalar curvature goes to zero at infinity, improving a result by Cao-Cui. Secondly, we continue our investigation on steady soliton with fast curvature decay in complex geometric setting. We classify Kaehler steady gradient Ricci soliton with nonnegative Ricci curvature and integrable scalar curvature. It is shown that such a soliton is isometric to a quotient of a product of the cigar soliton and some Kaehler Ricci flat manifold. As an application, we classify Kaehler steady gradient Ricci soliton with nonnegative Ricci curvature and scalar curvature decay rate faster than linear rate. Thirdly, we study the scalar curvature of noncompact expanding Ricci soliton. We derive a sharp lower bound for the scalar curvature of noncompact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We then give a sufficient condition for the scalar curvature of expanding gradient soliton being nonnegative. As an application, we prove that a three dimensional noncompact expanding gradient Ricci soliton with scalar curvature decaying faster than quadratic rate must be flat.