Structure at infinity for shrinking Ricci solitons

- This paper concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder R x S"_1/r occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic at infinity to the same round cylinder. This result is applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It was previously known that such solitons with scalar curvature approaching zero at infinity must be smoothly asymptotic to a cone. For the case that the scalar curvature is bounded from below by a positive constant, we conclude that along each end the soliton is asymptotic to a quotient of R x S³ or converges to a quotient of R² x S² along each integral curve of the gradient vector field of the potential function.