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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