On Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

In this talk, we will discuss the geometry of 4-dimensional compact gradient Ricci solitons. We will show that, under an upper bound condition on the range of the potential function, a 4-dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates will be presented. In particular, we discuss some related open problems.
This is a joint work with Xu Cheng and Detang Zhou.