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

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.

Solved by gpt-5.6-sol high.