Laplacian of a restricted ambient function

If $M\subseteq {\mathbb{R}}^{n+m}$ and $f$ is smooth in the ambient space, then

For the squared distance $r^2(x)=|x{|}^2$, a minimal $n$-submanifold satisfies ${\Delta }_M{r}^2=2n$. On a compact boundaryless manifold, $r^2$ attains a maximum where the Laplacian is nonpositive, a contradiction for $n>0$.