Four-dimensional shrinkers with nonnegative Ricci curvature

TL;DR

This paper classifies 4-dimensional complete noncompact shrinkers with nonnegative Ricci curvature, identifying conditions under which they are isometric to products of Euclidean space and spheres, based on curvature bounds and asymptotic behavior.

Contribution

It provides new classification results for 4D shrinkers with nonnegative Ricci curvature, linking curvature bounds to geometric structures like  and  products.

Findings

Shrinkers with sectional curvature  are isometric to  .

Scalar curvature  bounds imply topological and geometric rigidity.

Conditions on Ricci eigenvalues lead to classification as  .

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 $\chi(M)\geq 0$ 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 $R\le\tfrac{3}{2}-\delta$ for some $\delta\in (0,\tfrac{1}{2}]$, 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.