A gap theorem on complete shrinking gradient Ricci solitons

Abstract

In this short note, using G\"unther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature $K(g)<A$ and ${\rm Vol}_{f}(M)\geq v$ for some uniform constant $A,v$, there exists a small uniform constant $\epsilon_{n,A,v}>0$ depends only on $n, A$ and $v$, if the scalar curvature $R\leq \epsilon_{n,A,v}$, then $(M,g,f)$ is isometric to the Gaussian soliton $(\mathbb{R}^{n}, g_{E}, \frac{|x|^{2}}{4})$.