 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 $\mathbf{H}=0$, soCompactness 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.

Solved by gpt-5.6-sol high.