Convexity of $\lambda$-hypersurfaces

We prove that any $n$-dimensional closed mean convex $\lambda$-hypersurface is convex if $\lambda \le 0.$ This generalizes Guang’s work on $2$-dimensional strictly mean convex $\lambda$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $\lambda$-hypersurfaces with $\lambda \le 0.$