The rigidity theorems for complete self-expander of mean curvature flow

Our first main result is a rigidity theorem for complete self-expanders in the Euclidean space with higher codimension, assuming an integral cur More precisely, any smooth complete self-expander $x:M\to \mathbb R^{n+p}$($n\geq3$) that satisfies both $(\int_{M}|A|^{n}dμ)^{\frac{1}{n}}<K(n)$ and $\int_{M}|A|^{n}e^{\frac{|x|^{2}}{2}}dμ<\infty$ for a positive constant $K(n)$ depending only o isometric to $\mathbb R^{n}$. Moreover, we show that the rigidity result persists when the pinching condition is expressed in terms of the trace-free second fundamental form.