On Shrinking 4-Solitons with Sectional Curvature Bounded Above

To understand the geometric properties of shrinking gradient Ricci solitons, is an important step as one using the Ricci flow theory to solve the differential geometric problems. In this paper, the author studies the geometric properties of shrinking 4-solitons by using the standard maximum principle, and obtains an important curvature estimate. More precisely, on a compact shrinking 4-soliton, if the sectional curvature has a suitable upper bound, then the Ricci curvature should be nonnegative. When the soliton is non-compact, if in addition, the scalar curvature is bounded and has a positive lower bound, then a similar result will hold. In particular, the upper bound of the sectional curvature is optimal.