Preface The motivation for these lecture notes on minimal surfaces is to have a treatment that begins with almost no prerequisites and ends up with current research topics. We touch upon some of the applications to other fields including low dimensional topology, general relativity, and materials science. Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the max- imum principle arising in Bernstein’s classical work, and even Lebesgue’s definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), we have chosen to focus on the PDE aspects of the theory. In Chapter 1, 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 first chapter is on the 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 of Chapter 1, 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 ix
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