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Minimal surfaces. (English) Zbl 0987.49025

Courant Lecture Notes in Mathematics. 4. New York, NY: Courant Institute of Mathematical Sciences. viii, 124 p. (1999).
From the preface: “These notes are an expanded version of a one-semester course taught at Courant Institute in the spring of 1998. The only prerequisites needed 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 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 last section of Chapter 1, we derive the second variation formula, the stability inequality, and define the Morse index of a minimal surface.
Chapter 2 deals with generalizations of the Bernstein theorem discussed in Chapter 1. We begin the chapter by deriving Simons’ inequality for the Laplacian of the norm squared of the second fundamental form of a minimal hypersurface \(\Sigma\) in \(\mathbb R^n\). In the later sections, we discuss various applications of such an inequality. The first application that we give is to a theorem of Choi-Schoen giving curvature estimates for minimal surfaces with small total curvature. Using this estimate, we give a short proof of Heinz’ curvature estimate for minimal graphs. Next, we discuss a priori estimates for stable minimal surfaces in three-manifolds, including estimates on area and total curvature of Colding-Minicozzi and the curvature estimate of Schoen. After that, we follow Schoen-Simon-Yau and combine Simons’ inequality with the stability inequality to show higher \(L^p\) bounds for the square of the norm of the second fundamental form for stable minimal hypersurfaces. The higher \(L^p\) bounds are then used together with Simons’ inequality to show curvature estimates for stable minimal hypersurfaces and to give a generalization due to De Giorgi, Almgren, and Simons of the Bernstein theorem proven in Chapter 1. We close the chapter with a discussion of minimal cones in Euclidean space and the relationship to the Bernstein theorem.
We start Chapter 3 by introducing stationary varifolds as a generalization of classical minimal surfaces. After that, we prove a generalization of the Bernstein theorem for minimal surfaces discussed in the preceding chapter. Namely, following [T. H. Colding and W. P. Minicozzi II, Commun. Pure Appl. Math. 51, No. 2, 113-138 (1998; Zbl 0928.58022)], we show in Chapter 3 that, in fact, a bound on the density gives an upper bound for the smallest affine subspace that the minimal surface lies in. We deduce this theorem from the properties of the coordinate functions (in fact, more generally properties of harmonic functions) on \(k\)-rectifiable stationary varifolds of arbitrary codimension in Euclidean space.
Chapter 4 discusses the solution to the classical Plateau problem, focusing primarily on its regularity. The first three sections cover the basic existence result for minimal disks. After some general discussion of unique continuation and nodal sets, we study the local description of minimal surfaces in a neighborhood of either a branch point or a point of nontransverse intersection. Following Osserman and Gulliver, we rule out interior branch points for solutions of the Plateau problem. In the remainder of the chapter, we prove the embeddedness of the solution to the Plateau problem when the boundary is in the boundary of a mean convex domain. This last result is due to Meeks and Yau.
Finally, in Chapter 5, we discuss the theory of minimal surfaces in three-manifolds. We begin by explaining how to extend the earlier results to this case (in particular, monotonicity, the strong maximum principle, and some of the other basic estimates for minimal surfaces). Next, we prove the compactness theorem of Choi and Schoen for embedded minimal surfaces in three-manifolds with positive Ricci curvature. An important point for this compactness result is that by results of Choi-Wang and Yang-Yau such minimal surfaces have uniform area bounds”.

MSC:

49Q05 Minimal surfaces and optimization
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)

Citations:

Zbl 0928.58022
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