Use the convention
$$
A(X,Y)=(\overline\nabla_XY)^\perp,
\qquad
\mathbf H=\sum_{i=1}^nA(e_i,e_i).
$$
An embedded submanifold is <minimal surface>[minimal] when its mean curvature vector $\mathbf H$ vanishes identically. For a variation $M_t$ with velocity field $V$, the <first variation of area> is
$$
\left.\frac d{dt}\operatorname{Area}(M_t)\right|_{t=0}
=-\int_M\langle\mathbf H,V\rangle,d\mu
+\int_{\partial M}\langle V,\eta\rangle,d\sigma,
$$
where $\eta$ 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.

Solution

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

{wiki=First_variation_of_area_formula}

With $A(X,Y)=(\overline\nabla_XY)^\perp$ and $\mathbf H=\operatorname{tr}A$, a variation with velocity $V$ satisfies
$$
\left.\frac d{dt}\operatorname{Area}(M_t)\right|_{t=0}
=-\int_M\langle\mathbf H,V\rangle,d\mu
+\int_{\partial M}\langle V,\eta\rangle,d\sigma.
$$
Thus a boundaryless submanifold is stationary for every compactly supported variation exactly when it is minimal.


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 Incoming links: First variation of area