Geometry of 3-Dimensional Gradient Ricci Solitons with Positive Curvature

In this article, we mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. Throughout this paper, we denote such solitons by (M3,g). This paper is organized as follows. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of f. These estimates are crucial in the study of the diameter of level sets and the geometry of such solitons at infinity. In section 3, we show that the scalar curvature of such solitons approaches zero at infinity. The vanishing of the scalar curvature at infinity plays a significant role in Hamilton's dimension reduction argument on odd-dimensional gradient solitons. In section 4, we first investigate the growth rate of the diameter of level sets, which implies that the tangent cone of such solitons is a ray. Next we show that the scalar curvature falls off like 1/s and the diameter of geodesic spheres grows like square root s provided that R x Sigma-2 cannot occur as a limit from dimension reduction. Note that Perelman's no local collapsing theorem implies that R x Sigma-2 cannot be a limit of dilations about a finite time singularity. Therefore, if a gradient soliton (M3,g) arises as a limit of dilations of a compact solution, then R x Sigma-2 cannot occur as a limit from some dimension reduction since a limit of a limit is again a limit.

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Published 1 January 2005