Geometry of 3-Dimensional Gradient Ricci Solitons with Positive Curvature

Abstract

We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature of such solitons approaches zero at infinity. In section 4, we investigate the geometry of such solitons at infinity, e.g., the tangent cone, the asymptotic behavior, etc.