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Notes on Minimal Surfaces Notes on Minimal Surfaces Michael Beeson Aug. 9, 2007 with revisions and additions fall 2010 file last touched November 11, 2015 2 Contents 1 Introduction 5 1.1 NotationandBasicConcepts . 5 1.2 Weingartenmapandfundamentalforms . 6 1.3 MeanCurvatureandGaussCurvature . 8 1.4 Thedefinitionofminimalsurface . 10 1.5 Non-parametricminimalsurfaces . 12 1.6 Examplesofminimalsurfaces . 13 1.7 Calculusreview............................ 14 2 Harmonic Functions 17 2.1 Complex differentials . 17 2.2 ThePoissonintegral ......................... 20 2.3 The maximum principle . 22 2.4 Poisson’s formula in the upper half-plane . 23 2.5 Poisson formula for the half-plane, reprise . 26 2.6 Harmonic functions and Dirichlet’s integral . 28 3 Harmonic Surfaces 33 3.1 Isothermalcoordinates. 33 3.2 Uniformization ............................ 35 3.3 Minimal surfacesas harmonic conformalsurfaces . 36 3.4 Somegeometriccorollaries. 37 3.5 TheWeierstrassrepresentation . 42 3.6 Branchpoints............................. 43 4 Dirichlet’s Integral and Plateau’s Problem 45 4.1 TheDirichletIntegral ........................ 45 4.2 Dirichlet’s integral and area . 47 4.3 Plateau’sProblem .......................... 48 5 The Second Variation of Area 55 5.1 Thesecondvariationofareadefined . 55 5.2 The second variation of area for a normal variation . 57 5.3 The second variation of area is intrinsic . 59 3 4 CONTENTS 5.4 Non-normalvariations . .. .. .. .. .. .. .. .. .. .. 60 6 EigenvaluesandtheGaussMap 63 6.1 TheGaussmapofaminimalsurface . 63 6.2 Eigenvalueproblems ......................... 64 6.3 EigenvaluesandtheGaussMap. 66 6.4 The eigenvalue problem associated with D2A[u].......... 66 6.5 TheGauss-Bonnettheorem . 69 6.6 The Gauss-Bonnet theorem for branched minimal surfaces . 70 6.7 LaplacianoftheGaussmapofaminimalsurface . 71 7 Second Variation of Dirichlet’s Integral 73 7.1 Tangentvectorsandtheweakinnerproduct. 73 7.2 Theconformalgroup......................... 74 7.3 Calculation of the second variation of E .............. 75 7.4 Second variation of E andcomplexanalysis . 77 7.5 ForcedJacobifields.......................... 80 8 Dirichlet’s Integral and Area 83 8.1 From the kernel of D2E to the kernel of D2A ........... 83 8.2 From the kernel of D2A to the kernel of D2E ........... 84 9 SomeTheoremsofTomiandB¨ohme 93 9.1 Tomi’sno-immersed-loopstheorem . 93 9.2 B¨ohmeandTomi’sstructuretheorem . 96 9.3 Tomi’s Finiteness Theorem . 97 Chapter 1 Introduction These lectures contain some of the fundamental results needed to read papers on minimal surfaces. I have attempted to make them as self-contained as possible, rather than just a list of references to books containing these results. The title doesn’t mention the word “Introduction”, although the background assumed here is only calculus and the most elementary parts of the theory of functions of a complex variable. The necessary differential geometry and theory of harmonic functions is introduced and proved. However, a true introduction to minimal surfaces would involve more pictures of examples, and discussion of other results not presented here. For such books, see the list of references. These lectures have a different purpose: to supply proofs that don’t constantly refer you to some other place for the details. At present that aim has not been completely achieved; for example Lichtenstein’s theorem is not completely proved here. The first chapters accompanied lectures given at San Jos´ein November 2001. Later chapters were added in July, 2007, including the unpublished material from [2]. The bibliography lists a few reference books on the subject. There are many more, which explore different aspects of the theory of minimal surfaces. 1.1 Notation and Basic Concepts The open unit disk is D; the closed unit disk is D¯; the unit circle is S1. Cn means possessing n continuous derivatives. C0 means continuous. Surfaces of disk type are given by maps u : D¯ R3. We will suppose they are at least C3 in D and C1 on the boundary. Partial7→ derivatives will be denoted by subscripts ux and uy. A surface is regular at a point (x, y) in D if the tangent plane is well-defined there, i.e. ux and uy have nonzero cross product. Surfaces are required to be regular except at isolated points. A “regular point” of u is a point where u is regular. Another way of expressing regularity is that the Jacobian matrix u = u ,u has maximal rank two. ∇ h x yi A Jordan curve is a continuous one-one map Γ from S1 into R3. A reparametriza- tion of Γ is another Jordan curve of the form Γ φ, where φ is a one-one map ◦ 5 6 CHAPTER 1. INTRODUCTION of S1 into S1. A surface u is said to be bounded by Γ in case u restricted to S1 is a reparametrization of Γ. Plateau’s Problem is this: Given a Jordan curve Γ, find a surface of least area bounded by Γ. The space of all vectors in R3 which are tangent to the surface u at a regular point (x, y) is a vector space Tp, called the tangent space. A basis for the space is formed by ux and uy. The unit normal N = N(x, y) at p is given by u u N = x × y u u | x × y| We claim that N and N are tangent vectors. Proof : N N = 1. Dif- x y · ferentiating, we have N Nx = 0 and N Ny = 0, so Nx and Ny are tangent vectors. · · 1.2 Weingarten map and fundamental forms In this section, we fix a point (x, y) at which u is regular, i.e. ux uy does not vanish. We have assumed that non-regular points of a surface are× isolated, by definition. The Weingarten map S = S(x, y) is a linear map of Tp into itself, defined as 1 2 follows: If v = v ux + v uy then Sv = v1N v2N = viN − x − y − i i where the repeated subscript implies summation, and v means vi, but consid- ered as a column vector (“contravariant”). (We always sum one raised index times one lowered index.) The letter S is used because this is also known as the “shape operator”. In this section we write ui for the derivative of u with respect to xi, risking confusion with the components of u, but avoiding double subscripts as in uxi . Lemma 1 The Weingarten map is self-adjoint. That means Sv w = v Sw. · · Proof : Differentiate N u = 0 with respect to x . We get · i j N u + N u =0. j · i · ij Therefore N u = N u i · j j · i since both are equal to N u . Hence · ij Sv w = N vi u wj · − i · j = N u viwj − i · j = u N viwj − i · j = u vi N wj − i · j = v Sw · 1.2. WEINGARTEN MAP AND FUNDAMENTAL FORMS 7 That completes the proof of the lemma. Thus we can define the following three symmetric bilinear forms: I(v, w) := v w · II(v, w) := Sv w · III(v, w) := Sv Sw · These are called the first fundamental form, second fundamental form, and third fundamental form of u. There are several systems of notation for the coefficients of the fundamental forms. We have g := u u ij i · j b := N u ij − i · j c := N N ij i · j Thus the bij are the entries in the matrix of the shape operator; that is why they are defined with a minus sign. Since N u = 0, by differentiating we have · j Ni uj + N uij =0, so · · b = u N ij ij · In some books this is taken as the definition of bij , and then the fact that the bij are the coefficients of the shape operator is a lemma, instead of the other way around as here. Remember bij = bji = N uij , and of course gij = gji. The older (nineteenth-century)· notation uses E, F, G, L, M, and N, defined by g g E F G := 11 12 = g21 g22 F G b b L M B := 11 12 = b21 b22 M N Hopefully, when you see this notation (in old books), you won’t confuse N with the unit normal. Here we have used a different font to make the distinction. We have 2 W = √g = EG F = det(gij ) p − q We define gij to be the coefficients of the inverse of G, so g11 g12 1 G F = − g21 g22 W 2 F E − The Weingarten equations tell how to compute Nx and Ny in terms of ux and uy: N = bj u i − i j 8 CHAPTER 1. INTRODUCTION j kj where bi = bikg , and as usual repeated indices imply summation. To find this j j j formula for the bi , first write, with unknown coefficients ai , Ni = ai uj . Now take the dot product with uk: N u = aj u u i · k i j · k That is, b = ajg − ik i jk j Now we can solve for the ai by using the inverse matrix of G: j kp j p ai gjkg = ai δj p = ai = b gkp − ik This is the formula for the coefficients in the Weingarten equations. 1.3 Mean Curvature and Gauss Curvature The geometric meaning of the second fundamental form can best be seen by finding an orthonormal basis in which the Weingarten map S is diagonal. Let κ1 and κ2 be the eigenvalues of the matrix of S. Since S is self-adjoint, standard linear algebra tells us that it can be diagonalized, and that the eigenvalues are the minimum and maximum of the Rayleigh quotient II(v, v)/I(v, v). Let p and q form a basis in which S is diagonal. If κ = κ then p and q are automatically 1 6 2 orthogonal, since κ1p q = Sp q = p Sq = κ2p q. If κ1 = κ2, however, then any orthogonal unit vectors· p and· q will· do.1 · Thinking geometrically instead of algebraically, we write pN = κ p instead ∇ 1 of the equivalent S(p) = κ1p.
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