I have decided to post all of my notes on minimal surfaces. These notes are essentially a summary of how I spent my summer. Each post will build on the previous and there will be four in total. 1 area functional, 2 harmonic function and isothermal coordinates, 3 Weierstrass-Enneper representation, 4 examples.

Definition. Let $\phi(x,y)$ be a real valued function of two real variables $x$ and $y$ defined on a domain $D$. The partial differential equation

$\displaystyle \Delta\phi:=\phi_{xx}(x,y)+\phi_{yy}(x,y)=0$

is known as Laplace’s equation. If $\phi$, $\phi_x$, $\phi_y$, $\phi_{xx}$, $\phi_{x,y}$, $\phi_{y,x}$, and $\phi_{yy}$ are all continuous and if $\phi(x,y)$ satisfies Laplace’s equation then $\phi(x,y)$ is harmonic on $D$.

An interesting relationship between minimal surfaces and harmonic functions comes about when the surface is parametrized by isothermal coordinates

Definition. A parameterization $\textbf{x}(u,v)$ is called isothermal if $E=\langle\textbf{x}_u,\textbf{x}_u\rangle=\langle\textbf{x}_v,\textbf{x}_v\rangle=G$ and $F=\langle\textbf{x}_u,\textbf{x}_v\rangle=0$

Theorem. Isothermal coordinates exist on any surface $M\subseteq\mathbb{R}^3$.

Proof. A Survey of Minimal Surfaces [Osserman]. $\Box$

When isothermal parameters are used, there is a close relationship between the Laplace operator $\Delta\textbf{x}=\textbf{x}_{uu}+\textbf{x}_{vv}$ and mean curvature. We have the following formulas for an orthogonal coordinate system

$\textbf{x}_{uu}=\frac{E_u}{2E}\textbf{x}_u-\frac{E_v}{2G}\textbf{x}_v+lU$

$\textbf{x}_{uv}=\frac{E_v}{2E}\textbf{x}_u-\frac{G_u}{2G}\textbf{x}_v+mU$

$\textbf{x}_{vv}=-\frac{G_u}{2E}\textbf{x}_u+\frac{G_v}{2G}\textbf{x}_v+nU.$

Theorem. If the patch $\textbf{x}(u,v)$ is isothermal then $\Delta\textbf{x}(u,v)=\textbf{x}_{uu}+\textbf{x}_{vv}=(2EH)U$.

Proof. Since $E=G$ and $F=0$, we have

$= \textbf{x}_{uu}+\textbf{x}_{vv}$

$= \left(\frac{E_u}{2E}\textbf{x}_u-\frac{E_v}{2G}\textbf{x}_v+lU\right)+\left(-\frac{G_u}{2E}\textbf{x}_u+\frac{G_v}{2G}\textbf{x}_v+nU\right)$

$= \frac{E_u}{2E}\textbf{x}_u-\frac{E_v}{2G}\textbf{x}_v+lU-\frac{E_u}{2E}\textbf{x}_u+\frac{E_v}{2E}\textbf{x}_v+nU$

$= (l+n)U$

$= 2E\left(\frac{l+n}{2E}\right)U.$

By examining the formula for mean curvature when $E=G$ and $F=0$, we see that

$\displaystyle H=\frac{lG-2mF+nE}{2EG-F^2}=\frac{lE+nE}{2E^2}=\frac{E(l+n)}{2E^2}=\frac{l+n}{2E}.$

Therefore, $\textbf{x}_{uu}+\textbf{x}_{vv}=(2EH)U$. $\Box$

Corollary. A surface $M:\textbf{x}(u,v)=\left(x^1(u,v),x^2(u,v),x^3(u,v)\right)$ with isothermal coordinates is minimal if, and only if, $x^1$, $x^2$, and $x^3$ are harmonic functions.

Proof. If $M$ is minimal then $H=0$ and, by the previous theorem, $\textbf{x}_{uu}+\textbf{x}_{vv}=0.$ On the other hand, suppose that $x^1$, $x^2$, and $x^3$ are harmonic functions. Then $\textbf{x}(u,v)$ is harmonic so $\textbf{x}_{uu}+\textbf{x}_{vv}=0$ and, by the previous theorem, $(2EH)U=0$. Therefore, since $U$ is the unit normal and $E=\langle\textbf{x}_u,\textbf{x}_u\rangle\neq 0$, then $H=0$ and $M$ is minimal. $\Box$