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

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