= Solution
Suppose that a positive-dimensional compact boundaryless minimal submanifold $M^n\subseteq\mathbb R^{n+m}$ existed. Apply part b to the squared ambient distance $f(x)=|x|^2$. Its ambient Hessian is $2$ times the Euclidean metric, its gradient is $2x$, and $\mathbf H=0$, so
$$
\Delta_M|x|^2=2n.
$$
Compactness makes $|x|^2$ attain a maximum. The <Laplacian at a local maximum> is nonpositive, contradicting $2n>0$. Hence no such compact Euclidean minimal submanifold exists.
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