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 be a real valued function of two real variables and defined on a domain . The partial differential equation

is known as *Laplace’s equation*. If , , , , , , and are all continuous and if satisfies Laplace’s equation then is *harmonic* on .

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

**Definition.** A parameterization is called *isothermal* if and

**Theorem.** Isothermal coordinates exist on any surface .

*Proof. A Survey of Minimal Surfaces *[Osserman].

When isothermal parameters are used, there is a close relationship between the Laplace operator and mean curvature. We have the following formulas for an orthogonal coordinate system

**Theorem.** If the patch is isothermal then .

*Proof.* Since and , we have

By examining the formula for mean curvature when and , we see that

Therefore, .

**Corollary.** A surface with isothermal coordinates is minimal if, and only if, , , and are harmonic functions.

*Proof.* If is minimal then and, by the previous theorem, On the other hand, suppose that , , and are harmonic functions. Then is harmonic so and, by the previous theorem, . Therefore, since is the unit normal and , then and is minimal.

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