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
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