Sun-Chin Chu

Noncompact gradient Ricci solitons

There is no elite Just take your place in the drivers seat. – From Drivers Seat by Sniff n the Tears Gradient Ricci solitons (GRS), which were introduced and used effectively in the Ricci flow by Hamilton, are generalizations of Einstein metrics. A motivation for studying GRS is that they arise in the analysis of singular solutions. By Myers theorem, there are no noncompact Einstein solutions to the Ricci flow with positive scalar curvature (which would homothetically shrink under the Ricci flow). In view of this, regarding noncompact GRS, one may expect to obtain the most information in the shrinking case. As we shall see in this chapter, this appears to be true. A beautiful aspect of the study of GRS is the duality between the metric and the potential function (we use the term duality in a non-technical way). On one hand, associated to the metric are geodesics and curvature. On the other hand, associated to the potential function are its gradient and Laplacian as well as its level sets and the integral curves of its gradient. In this chapter we shall see some of the interaction between quantities associated to the metric and to the potential function, which yield information about the geometry of GRS. In Chapter 1 of Part I we constructed the Bryant soliton and we discussed some basic equations holding for GRS, leading to a no nontrivial steady or expanding compact breathers result. In the present chapter we focus on the qualitative aspects of the geometry of noncompact GRS. In §1 we discuss a sharp lower bound for their scalar curvatures. In §2 we present estimates of the potential function and its gradient for GRS. In §3 we improve some lower bounds for the scalar curvatures of nontrivial GRS. In §4 we show that the volume growth of a shrinking GRS is at most Eu-clidean. If the scalar curvature has a positive lower bound, then one obtains a stronger estimate for the volume growth. In §5 we discuss the logarithmic Sobolev inequality on shrinking GRS. In §6 we prove that shrinking GRS with nonnegative Ricci curvature must have scalar curvature bounded below by a positive constant. Although much is known about GRS, there is still quite a lot that is unknown. In this chapter we include some problems and conjectures (often