Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 2

The metric and orientation determine the Riemannian volume form: in a positively oriented coordinate chart $(x^1,\dots ,x^n)$,Writing $|g|=\det (g_{ij})$ and $(g^{ij})=(g_{ij}{)}^{-1}$, the Dirichlet energy on a Riemannian manifold with source $f$ is

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For an arbitrary smooth variation $u_t=u+t\phi$, differentiation under the integral givesSince $M$ is compact without boundary, integration by parts turns this intoThe fundamental lemma of the calculus of variations therefore gives the Euler-Lagrange equationor equivalently $-{\Delta }_{g}u=f$ for the Laplace-Beltrami operator.

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Integrating the Euler-Lagrange equation and applying the divergence theorem on the compact boundaryless manifold givesEquivalently, if this integral were nonzero, replacing $u$ by $u+C$ would leave the gradient term unchanged and make the energy unbounded below in one direction, so no minimizer could exist.

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