Chapter 1

The Beginning of the

Theory

In this chapter, we will first derive the minimal surface equation as the

Euler-Lagrange equation for the area functional on graphs. Subsequently,

we derive the parametric form of the minimal surface equation (the first

variation formula). The focus of the chapter is on some basic properties

of minimal surfaces, including the monotonicity formula for area and the

Bernstein theorem. We also mention some examples. In the next to last

section, we derive the second variation formula, the stability inequality, and

define the Morse index of a minimal surface. In the last section, we introduce

multi-valued minimal graphs which will play a major role later when we

discuss results from [CM3]–[CM7]. We will also give a local example,

from [CM18], of spiraling minimal surfaces (like the helicoid) that can be

decomposed into multi-valued graphs, but where the rate of spiraling is far

from constant.

1. The Minimal Surface Equation and Minimal

Submanifolds

1.1. Graphs and the minimal surface equation. Suppose that u : Ω ⊂

R2

→ R is a

C2

function and consider the graph of the function u

(1.1) Graphu = {(x, y, u(x, y)) | (x, y) ∈ Ω} .

Then the area is

Area(Graphu) =

Ω

|(1, 0,ux) × (0,1,uy)| (1.2)

=

Ω

1 + ux 2 + uy 2 =

Ω

1 + |∇u|2 ,

1