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$M^n\subset \mathbb R^{n+1}$ is a compact without boundary n-dimensional smooth manifold. I see a first variation formula $$ \int_M div _M Y = -\int_M \langle H , Y \rangle $$ $Y$ is a vector field on $M$, $H$ is mean curvature vector. If $Y$ is a tangent vector field , it is obvious right. But if $Y$ have normal component, first, how to define the $div_M Y$. Second, how to show the formula ?

Enhao Lan
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1 Answers1

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During the proof of the first variation formulae, for a isometric immersion $\psi: M^n \to N^{n+k}$, you have that

$$\frac{d}{dt}dM_t \bigg|_{t=0}= -n\langle H,V\rangle + div(V^T),$$

where $V$ is the variation field of $\psi$. Then the term of the dirvegence is well defined.

Irddo
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  • Thanks, where I can find the proof of the first variation formula? Because I don't know how to get $\frac{d}{dt}dM_t \bigg|_{t=0}= -n\langle H,V\rangle + div(V^T)$. – Enhao Lan May 18 '17 at 13:19
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    Take a look at Colding & Minicozzi book's: A course of Minimal Surfaces. – Irddo May 18 '17 at 14:42