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 $x:M\to {\mathbb{R}}^{n+p}$($n\ge 3$) that satisfies both $({\int }_M\mathrm{\mid }A{\mathrm{\mid }}^{n}d\mu {)}^{\frac{1}{n}}<K(n)$ and ${\int }_M\mathrm{\mid }A{\mathrm{\mid }}^n{e}^{\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{2}}d\mu <\mathrm{\infty }$ 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.