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

Use the conventionAn embedded submanifold is minimal when its mean curvature vector $\mathbf{H}$ vanishes identically. For a variation $M_t$ with velocity field $V$, the first variation of area iswhere $\eta$ is the outward unit conormal. For a boundaryless $M$, only the first integral remains. This sign convention is consistent with the outward variation of a round sphere increasing its area, because its $\mathbf{H}$ points inward.

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Let $\overline{\nabla }f$ and $\overline{\mathrm{Hess}}f$ be the ambient gradient and Hessian. The Laplacian of a restricted ambient function isTo prove it, choose a local orthonormal tangent frame $e_1,\dots ,e_n$ with ${\nabla }_{e_i}^M{e}_j=0$ at the point under consideration. There,Both sides are intrinsic scalars, so the pointwise calculation proves the formula everywhere.

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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.

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