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= First variation of area
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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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codex/first-variation-of-area

First variation of area formula

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First variation of area formula

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