A gap theorem on complete shrinking gradient Ricci solitons

Abstract:

In this short note, using Günther’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 $\textrm {Vol}_{f}(M)\geq v$ 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 $(\mathbb {R}^{n}, g_{E}, \frac {|x|^{2}}{4})$.