Ngô Quốc Anh

February 8, 2011

Minimal Surfaces: The mean curvature condition

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:21

Suppose that $\vec x(u,v)$, $(u,v) \in I_1 \times I_2$ is the surface with minimal area among those whose boundary coincides with that of $\vec x$. Let $t(u,v)$ be any smooth function on $I_1 \times I_2$ such that it vanishes on the boundary of $\vec x$

Consider the following family of regular parametrized surfaces

${\vec x}^\varepsilon = \vec x(u,v)+\varepsilon t(u,v)N, \quad -\lambda < \varepsilon < \lambda$

for some small $\lambda$. These surfaces can be thought of as being obtained by varying $\vec x(u,v)$ along its normals, while fixing its boundary. Note that

$\displaystyle {E^\varepsilon } = \langle \vec x_u^\varepsilon ,\vec x_u^\varepsilon \rangle = E + 2\left\langle {{{\vec x}_u},\varepsilon \left( {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right)} \right\rangle + {\varepsilon ^2}{\left\| {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right\|^2}$

and

$\displaystyle \begin{gathered} {F^\varepsilon } = \langle \vec x_u^\varepsilon ,\vec x_v^\varepsilon \rangle \hfill \\ \quad\;= F + \left\langle {{{\vec x}_u},\varepsilon \left( {\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right)} \right\rangle + \left\langle {{{\vec x}_v},\varepsilon \left( {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right)} \right\rangle + {\varepsilon ^2}\left\langle {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}},\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right\rangle \hfill \\ \end{gathered}$

and

$\displaystyle {G^\varepsilon } = \langle \vec x_v^\varepsilon ,\vec x_v^\varepsilon \rangle = G + 2\left\langle {{{\vec x}_v},\varepsilon \left( {\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right)} \right\rangle + {\varepsilon ^2}{\left\| {\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right\|^2}$

Let

$A(\varepsilon)={\rm area \; of \;} {\vec x}^\varepsilon$.

If ${\vec x}^0=\vec x$ achieves minimal area, then

$\displaystyle \frac{d}{d\varepsilon}\bigg|_{\varepsilon=0} A(\varepsilon)=0.$

So

$\displaystyle\begin{gathered} 0 = {\left. {\frac{d}{{d\varepsilon }}} \right|_{\varepsilon = 0}}\int_{{I_2}} {\int_{{I_1}} {\sqrt {{E^\varepsilon }{G^\varepsilon } - {{({F^\varepsilon })}^2}} } } dudv \hfill \\ \quad= \int_{{I_2}} {\int_{{I_1}} {{{\left. {\frac{d}{{d\varepsilon }}} \right|}_{\varepsilon = 0}}} } \sqrt {{E^\varepsilon }{G^\varepsilon } - {{({F^\varepsilon })}^2}} dudv \hfill \\ \quad= \int_{{I_2}} {\int_{{I_1}} {\frac{1}{{2\sqrt {EG - {F^2}} }}} } \left( {2E\left\langle {{{\vec x}_v},\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right\rangle + 2G\left\langle {{{\vec x}_u},\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right\rangle - } \right. \hfill \\ \qquad\left. {2F\left( {\left\langle {{{\vec x}_u},\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right\rangle + \left\langle {{{\vec x}_v},\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right\rangle } \right)} \right)dudv \hfill \\ \quad= \int_{{I_2}} {\int_{{I_1}} {\frac{t}{{\sqrt {EG - {F^2}} }}} } ( - E\langle N,{{\vec x}_{vv}}\rangle - G\langle N,{{\vec x}_{uu}}\rangle + 2F\langle N,{{\vec x}_{uv}}\rangle )dudv \hfill \\ \quad= - \int_{{I_2}} {\int_{{I_1}} 2 } tH\sqrt {EG - {F^2}} dudv. \hfill \\ \end{gathered}$

If we take

$t(u,v)=H(u,v)$

then we have

$\displaystyle - \int_{{I_2}} {\int_{{I_1}} 2 } {H^2}\sqrt {EG - {F^2}} dudv = 0.$

Note that

$2H^2\sqrt{EG-F^2} \geqslant 0$

and

$\sqrt{EG-F^2}>0$

(as $\sqrt{EG-F^2}=\|\vec x_u \times \vec x_v\| \ne 0$, if $\vec x$ is regular). So

$H(u,v)=0, \quad \forall (u,v) \in I_1 \times I_2$

which is precisely the necessary condition we want.

Definition. A smooth surface with vanishing mean curvature is called a minimal surface.

Example 1. Let $\vec x(u,v)=(u,v,f(u,v))$ be the graph of $f(u,v)$, a smooth function on $(u,v) \in I_1 \times I_2$. Then $\vec x$ is a minimal surface if

$\displaystyle \left( 1+\left(\frac{\partial f}{\partial u}\right)^2\right)\frac{\partial^2f}{\partial v^2} + \left( 1+\left(\frac{\partial f}{\partial v}\right)^2\right)\frac{\partial^2 f}{\partial u^2}-2\frac{\partial f}{\partial u}\frac{\partial f}{\partial v}\frac{\partial^2 f}{\partial u \partial v}=0.$

The above equation is called the minimal surface equation.

Example 2. Let us consider the so-called Enneper surface given by

$\displaystyle\vec{x}(u, v)=\left(u-\frac{1}{3}u^3+uv^2, -v+\frac{1}{3}v^3-vu^2, u^2-v^2\right).$

Then it is known that

$\displaystyle {{\vec x}_u} = (1 - {u^2} + {v^2}, - 2uv,2u),{{\vec x}_v} = (2uv, - 1 + {v^2} - {u^2}, - 2v)$

which implies

$\displaystyle {{\vec x}_u} \times {{\vec x}_v} = (2u({u^2} + {v^2} + 1),2v({u^2} + {v^2} + 1),({u^2} + {v^2} - 1)({u^2} + {v^2} + 1)).$

Thus

$\displaystyle \|{{\vec x}_u} \times {{\vec x}_v}\| = {({u^2} + {v^2} + 1)^2}N(u,v) = \left( {\frac{{2u}}{{{u^2} + {v^2} + 1}},\frac{{2v}}{{{u^2} + {v^2} + 1}},\frac{{{u^2} + {v^2} - 1}}{{{u^2} + {v^2} + 1}}} \right).$

It is easy to check that

$E=G=(u^2+v^2+1)^2, \quad F=0$

and

$e= \langle N,{{\vec x}_{uu}}\rangle=-2, \quad f=\langle N,{{\vec x}_{uv}}\rangle=0, \quad g=\langle N,{{\vec x}_{vv}}\rangle=2$

Hence $H=0$, and Enneper surface is a minimal surface.

Example 3. The catenoid surface given by

$\displaystyle\vec{x}(u, v)=(\cosh u\cos v, \cosh u\sin v, u)$

is also a minimal surface.