On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

Abstract

For all dimensions $n\ge 5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that ${\nabla }^{2}f$ is $2-$nonnegative and the curvature tensor is WPIC1 at some point $\bar{x}\in M$. Then $(M,g)$ must be a quotient of either $S^n$ or $S^{n-1}\times \mathbb{R}$. Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions $n\ge 9$. Namely that a complete ancient solution to the Ricci flow of dimension $n\ge 9$ with uniformly PIC must be weakly PIC2.