Lemma (the first variation forula for area, particular case): Let $% X:D\subset %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$ be a local parametric surface, with $D$ relatively compact domains. Denote by $\xi $ smooth function on $D$ and by $n$ the unit normal vector field of $X$ and set $Y=\xi n$, so the first variation of the area of $X$ is given by $\delta _{\xi n}\left\vert X\right\vert =-2\int_{D}\xi HdA$

Definition1: under the same conditions as above, the volume of $X\left( D\right) ,$denote by $\left\vert CX\right\vert $ is $\frac{1}{3}% \int_{D}\left\langle X,n\right\rangle dA$

Definition2: under the same conditions as above, consider $H_{0}=\frac{% \int_{D}HdA}{\left\vert X\right\vert }$, where $H$ is mean curvature of $X$, and define a functional $J_{D}\left( X\right) $ by $J_{D}\left( X\right) =\left\vert X\right\vert +2H_{0}\left\vert CX\right\vert $.

Now consider a variation of $X_{t}:D\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$, $t\in \left( -\varepsilon ,\varepsilon \right) $, with $X_{0}=X$

Let $H_{t}$ be the mean curvature of a surface $\left\{ D,X_{t}\right\} $ and $n_{t}$ the unit normal vector field of $X_{t}$, and put $\lambda _{t}=\left\langle \frac{\partial X_{t}}{\partial t},n_{t}\right\rangle $.

Consider $X_{t}$ instead of $X$ in $X_{t}=X+tY$, where $Y$ is vector field in $X\left( D\right) $

Show the details of the second equality

$J_{D}^{\prime }\left( X_{t}\right) =% \frac{d}{dt}J_{D}\left( X_{t}\right) =\int_{D}\left( -2H_{t}+2H_{0}\right) \lambda _{t}dM$

where $dM$ is the area element of $M$.

I know the $J_{D}^{\prime }\left( X_{0}\right) $, but I can not get the expression to $t$ any. I have searched in several books and articles, but people and general does not do this in detail.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.