Four-dimensional shrinkers with nonnegative Ricci curvature
Guoqiang Wu, Jia-yong Wu
TL;DR
The paper advances the classification of 4-dimensional, simply connected, complete noncompact shrinkers with nonnegative Ricci curvature by exploiting curvature pinching and precise asymptotics. It combines evolution estimates of Ricci-eigenvalues along integral curves of $\nabla f$, the Gauss-Bonnet-Chern formula with boundary terms, and barrier-based integration by parts to obtain rigidity results: under $K\le \tfrac{1}{4}$ or suitable bounds on $\lambda_1+\lambda_2$ with $R$ bounded, the soliton is $\mathbb{R}\times\mathbb{S}^3$; under $K\le \tfrac{1}{2}$ (or nonnegative bi-Ricci) with $R\le \tfrac{3}{2}-\delta$, it is $\mathbb{R}^2\times\mathbb{S}^2$; and when $R=1$ an alternative IBP-based proof yields the same cylinder structure. The work also provides an Euler-characteristic characterization in the $\mathbb{R}\times\mathbb{S}^3$ regime and develops barrier-method tools to extend divergence-type arguments to soliton geometry. Altogether, the paper sharpens the 4D shrinker landscape by linking asymptotic geometry, curvature pinching, and topological invariants under nonnegative Ricci curvature.
Abstract
In this paper, we investigate classifications of $4$-dimensional simply connected complete noncompact nonflat shrinkers satisfying $Ric+\mathrm{Hess}\,f=\tfrac 12g$ with nonnegative Ricci curvature. One one hand, we show that if the sectional curvature $K\le 1/4$ 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 $χ(M)\geq 0$ 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 $K\le 1/2$ (or the bi-Ricci curvature is nonnegative) and $R\le\tfrac{3}{2}-δ$ for some $δ\in (0,\tfrac{1}{2}]$, 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.