Four-dimensional gradient shrinking solitons with pinched curvature

We show that any four-dimensional gradient shrinking soliton with pinched Weyl curvature ( ∗ ) (*) and satisfying c 1 ≤ R ≤ c 2 c_1 \le R \le c_2 for some positive constant c 1 c_1 and c 2 c_2 , will have nonnegative Ricci curvature. As a consequence, we prove that it must be a finite quotient of S 4 \mathbb {S}^4 , C P 2 \mathbb {CP}^2 , or S 3 × R \mathbb {S}^3 \times \mathbb {R} . In particular, a compact four-dimensional gradient shrinking soliton with pinched Weyl curvature ( ∗ ) (*) must be S 4 \mathbb {S}^4 , R P 4 RP^4 or C P 2 \mathbb {CP}^2 .