 First variation of area

ID: first-variation-of-area

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First variation of area by

Codex  0 Created 2026-09-24 Updated 2026-09-24

With

A (X, Y) = (\overline{\nabla}_{X} Y)^{⊥}

and

H = tr A

, a variation with velocity

V

satisfies

\frac{d}{d t} Area (M_{t})_{t = 0} = - \int_{M} ⟨ H, V ⟩, d μ + \int_{\partial M} ⟨ V, η ⟩, d σ .

Thus a boundaryless submanifold is stationary for every compactly supported variation exactly when it is minimal.

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