Ends of Gradient Ricci Solitons

Self-similar solutions to Ricci flows, called Ricci solitons, are important geometric objects. To address the question whether new solitons can be constructed from existing ones through connected sums, we are led to investigate the issue of connectedness at infinity for solitons. The paper provides a brief account of our work along this line as well as a new result. The new result says that an n-dimensional gradient shrinking Ricci soliton is necessarily connected at infinity if its scalar curvature is bounded above by n3.