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.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 4 / c / Solution

If a twice differentiable real function has a local maximum at an interior point of a Riemannian manifold, its <Hessian matrix> is negative semidefinite there and hence its <Laplace-Beltrami operator>[Laplacian] is nonpositive.

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 Incoming links: Laplacian at a local maximum