UC San Diego

In the first three chapters, we study the steady gradient soliton, especially the 3-dimensional soliton with positive sectional curvatures and which is k-noncollapsed on all scales. In Chapter 1, we discuss the background of our project, and introduce the basic definitions. In Chapter 2, we get some geometric properties of the soliton. The asymptotic behaviors are studied. In Chapter 3, we introduce another approach looking at the difference between the two principle curvatures, and prove the soliton is rotationally symmetric out of a compact set under an additional assumption. In Chapter 4, we generalize Perelman's L-length to high dimensions. We define a new energy in a natural way, and derive the first variation of the energy. When the dimension is reduced to one, our first variation formula is exactly Perelman's L- geodesic equation. In Chapter 5 we study the mean curvature flow inside the Ricci flow. The interesting result is that in the evolution equation of the second fundamental form, many bad terms are canceled mysteriously due to the evolution of the ambient manifold