The constant-length second fundamental form conjecture for self-shrinkers

Let $\Sigma^n\subset\mathbb{R}^{n+1}$ be a smooth, complete, embedded self-shrinker, meaning that its mean curvature satisfies

$$H=\frac{\langle x,\mathbf{n}\rangle}{2}.$$

Assume that $\Sigma$ has polynomial volume growth and that the squared norm of its second fundamental form is constant. Constant-length second fundamental form conjecture. Then $\Sigma$ is a generalized cylinder.

This conjecture concerns the classification of self-shrinkers under a natural curvature constraint. It is solved: in dimension $n=2$, corresponding to self-shrinkers in $\mathbb{R}^3$, Ding and Xin proved the statement; the source presents a new proof of that result and establishes a gap theorem in all dimensions.