# A gap theorem on complete shrinking gradient Ricci solitons

## TL;DR

This paper proves a gap theorem for complete shrinking gradient Ricci solitons, showing that under certain curvature and volume conditions, a small scalar curvature implies the soliton is isometric to the Gaussian soliton.

## Contribution

It establishes a new gap theorem linking scalar curvature bounds to the Gaussian soliton structure under curvature and volume constraints.

## Key findings

- Scalar curvature bound implies Gaussian soliton under conditions
- Uses volume comparison and gap theorems for proof
- Provides explicit constants depending on dimension, curvature, and volume

## 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 $(\mathbb{R}^{n}, g_{E}, \frac{|x|^{2}}{4})$.