Four-dimensional shrinkers with nonnegative Ricci curvature

In this paper, we investigate classifications of $4$-dimensional simply connected complete noncompact nonflat shrinkers satisfying $Ric+\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}f=\frac{1}{2}g$ with nonnegative Ricci curvature. One one hand, we show that if the sectional curvature $K\le 1\mathrm{/}4$ or the sum of smallest two eigenvalues of Ricci curvature has a suitable lower bound, then the shrinker is isometric to $\mathbb{R}\times {\mathbb{S}}^3$. We also show that if the scalar curvature $R\le 3$ and the shrinker is asymptotic to $\mathbb{R}\times {\mathbb{S}}^3$, then the Euler characteristic $\chi (M)\ge 0$ and equality holds if and only if the shrinker is isometric to $\mathbb{R}\times {\mathbb{S}}^3$. On the other hand, we prove that if $K\le 1\mathrm{/}2$ (or the bi-Ricci curvature is nonnegative) and $R\le \frac{3}{2}-\delta$ for some $\delta \in (0,\frac{1}{2}]$, then the shrinker is isometric to ${\mathbb{R}}^2\times {\mathbb{S}}^2$. The proof of these classifications mainly depends on the asymptotic analysis by the evolution of eigenvalues of Ricci curvature, the Gauss-Bonnet-Chern formula with boundary and the integration by parts.