On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

Abstract

For all dimensions $n\geq5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that $\nabla^2f$ 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 $n\geq9$. Namely that a complete ancient solution to the Ricci flow of dimension $n\geq9$ with uniformly PIC must be weakly PIC2.