Integrating the Euler-Lagrange equation and applying the <divergence theorem> on the compact boundaryless manifold gives
$$
\int_M f\,d\operatorname{vol}_g
=-\int_M\Delta_gu\,d\operatorname{vol}_g=0.
$$
Equivalently, 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.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-115/2/c/solution