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Lecture Notes on Minimal Surfaces Lecture Notes on Minimal Surfaces Emma Carberry, Kai Fung, David Glasser, Michael Nagle, Nizam Ordulu February 17, 2005 Contents 1 Introduction 9 2 A Review on Differentiation 11 2.1 Differentiation........................... 11 2.2 Properties of Derivatives . 13 2.3 Partial Derivatives . 16 2.4 Derivatives............................. 17 3 Inverse Function Theorem 19 3.1 Partial Derivatives . 19 3.2 Derivatives............................. 20 3.3 The Inverse Function Theorem . 22 4 Implicit Function Theorem 23 4.1 ImplicitFunctions. 23 4.2 ParametricSurfaces. 24 5 First Fundamental Form 27 5.1 TangentPlanes .......................... 27 5.2 TheFirstFundamentalForm . 28 5.3 Area ................................ 29 5 6 Curves 33 6.1 Curves as a map from R to Rn .................. 33 6.2 ArcLengthandCurvature . 34 7 Tangent Planes 37 7.1 Tangent Planes; Differentials of Maps Between Surfaces . 37 7.1.1 Tangent Planes . 37 7.1.2 Coordinates of w T (S) in the Basis Associated to ∈ p Parameterization x .................... 38 7.1.3 Differential of a (Differentiable) Map Between Surfaces 39 7.1.4 Inverse Function Theorem . 42 7.2 TheGeometryofGaussMap . 42 7.2.1 Orientation of Surfaces . 42 7.2.2 GaussMap ........................ 43 7.2.3 Self-Adjoint Linear Maps and Quadratic Forms . 45 8 Gauss Map I 49 8.1 “Curvature”ofaSurface . 49 8.2 GaussMap ............................ 51 9 Gauss Map II 53 9.1 Mean and Gaussian Curvatures of Surfaces in R3 ....... 53 9.2 GaussMapinLocalCoordinates . 57 10 Introduction to Minimal Surfaces I 59 10.1 Calculating the Gauss Map using Coordinates . 59 10.2 MinimalSurfaces . 61 11 Introduction to Minimal Surface II 63 11.1 Why a Minimal Surface is Minimal (or Critical) . 63 11.2 ComplexFunctions . 66 11.3 Analytic Functions and the Cauchy-Riemann Equations . 67 6 11.4 HarmonicFunctions . 68 12 Review on Complex Analysis I 71 12.1 CutoffFunction .......................... 71 12.2 Power Series in Complex Plane . 73 12.3TaylorSeries............................ 76 12.3.1 The Exponential Functions . 77 12.3.2 The Trigonometric Functions . 78 12.3.3 TheLogarithm . 79 12.4 Analytic Functions in Regions . 79 12.5 ConformalMapping. 80 12.6 Zeros of Analytic Function . 81 13 Review on Complex Analysis II 83 13.1 Poles and Singularities . 83 14 Isotherman Parameters 87 15 Bernstein’s Theorem 93 15.1 Minimal Surfaces and isothermal parametrizations . 93 15.2 Bernstein’s Theorem: Some Preliminary Lemmas . 95 15.3 Bernstein’sTheorem . 98 16 Manifolds and Geodesics 101 16.1 ManifoldTheory . .101 16.2 PlateauProblem . .107 16.3Geodesics .............................108 16.4 CompleteSurfaces . .112 16.5 Riemannian Manifolds . 113 17 Complete Minimal Surfaces 115 17.1 CompleteSurfaces . .115 7 17.2 Relationship Between Conformal and Complex-Analytic Maps 117 17.3 RiemannSurface . .118 17.4 CoveringSurface . .121 18 Weierstrass-Enneper Representations 123 18.1 Weierstrass-Enneper Representations of Minimal Surfaces . 123 19 Gauss Maps and Minimal Surfaces 131 19.1 Two Definitions of Completeness . 131 19.2 Image of S undertheGaussmap . .132 19.3 Gauss curvature of minimal surfaces . 134 19.4 Complete manifolds that are isometric to compact manifolds minuspoints............................136 Bibliography 139 8 Chapter 1 Introduction Minimal surface has zero curvature at every point on the surface. Since a surface surrounded by a boundary is minimal if it is an area minimizer, the study of minimal surface has arised many interesting applications in other fields in science, such as soap films. In this book, we have included the lecture notes of a seminar course about minimal surfaces between September and December, 2004. The course consists of a review on multi-variable calculus (Lectures 2 - 4), some basic differential geometry (Lectures 5 - 10), and representation and properties of minimal surface (Lectures 11 - 16). The authors took turns in giving lectures. The target audience of the course was any advanced undergraduate student who had basic analysis and algebra knowledge. 9 Chapter 2 A Review on Differentiation Reading: Spivak pp. 15-34, or Rudin 211-220 2.1 Differentiation Recall from 18.01 that Definition 2.1.1. A function f : Rn Rm is differentiable at a Rn if → ∈ there exists a linear transformation λ : Rn Rm such that → f(a + h) f(a) λ(h) lim | − − | = 0 (2.1) h→0 h | | The norm in Equation 2.1 is essential since f(a + h) f(a) λ(h) is in Rm − − and h is in Rn. Theorem 2.1.2. If f : Rn Rm is differentiable at a Rn, then there is → ∈ a unique linear transformation λ : Rn Rm that satisfies Equation (2.1). → We denote λ to be Df(a) and call it the derivative of f at a Proof. Let µ : Rn Rm such that → f(a + h) f(a) µ(h) lim | − − | = 0 (2.2) h→0 h | | 11 and d(h)= f(a + h) f(a), then − λ(h) µ(h) λ(h) d(h)+ d(h) µ(h) lim | − | = lim | − − | (2.3) h→0 h h→0 h | | | | λ(h) d(h) d(h) µ(h) lim | − | + lim | − | (2.4) ≤ h→0 h h→0 h | | | | = 0. (2.5) Now let h = tx where t R and x Rn, then as t 0, tx 0. Thus, for ∈ ∈ → → x = 0, we have 6 λ(tx) µ(tx) λ(x) µ(x) lim | − | = | − | (2.6) t→0 tx x | | | | = 0 (2.7) Thus µ(x)= λ(x). Although we proved in Theorem 2.1.2 that if Df(a) exists, then it is unique. However, we still have not discovered a way to find it. All we can do at this moment is just by guessing, which will be illustrated in Example 1. Example 1. Let g : R2 R be a function defined by → g(x, y)=ln x (2.8) Proposition 2.1.3. Dg(a, b)= λ where λ satisfies 1 λ(x, y)= x (2.9) a · Proof. g(a + h, b + k) g(a, b) λ(h, k) ln(a + h) ln(a) 1 h lim | − − | = lim | − − a · | (h,k)→0 (h, k) (h,k)→0 (h, k) | | | | (2.10) 12 ′ 1 Since ln (a)= a , we have ln(a + h) ln(a) 1 h lim | − − a · | = 0 (2.11) h→0 h | | Since (h, k) h , we have | | ≥ | | ln(a + h) ln(a) 1 h lim | − − a · | = 0 (2.12) (h,k)→0 (h, k) | | Definition 2.1.4. The Jacobian matrix of f at a is the m n matrix of × Df(a) : Rn Rm with respect to the usual bases of Rn and Rm, and denoted → f ′(a). Example 2. Let g be the same as in Example 1, then 1 g′(a, b)=( , 0) (2.13) a Definition 2.1.5. A function f : Rn Rm is differentiable on A Rn if → ⊂ f is diffrentiable at a for all a A. On the other hand, if f : A Rm,A ∈ → ⊂ Rn, then f is called differentiable if f can be extended to a differentiable function on some open set containing A. 2.2 Properties of Derivatives Theorem 2.2.1. 1. If f : Rn Rm is a constant function, then a Rn, → ∀ ∈ Df(a) = 0. (2.14) 2. If f : Rn Rm is a linear transformation, then a Rn → ∀ ∈ Df(a)= f. (2.15) 13 Proof. The proofs are left to the readers Theorem 2.2.2. If g : R2 R is defined by g(x, y)= xy, then → Dg(a, b)(x, y)= bx + ay (2.16) In other words, g′(a, b)=(b, a) Proof. Substitute p and Dp into L.H.S. of Equation 2.1, we have g(a + h, b + k) g(a, b) Dg(a, b)(h, k) hk lim | − − | = lim | | (2.17) (h,k)→0 (h, k) (h,k)→0 (h, k) | | | | max( h 2, k 2) lim | | | | ≤ (h,k)→0 √h2 + k2 (2.18) √h2 + k2 (2.19) ≤ = 0 (2.20) Theorem 2.2.3. If f : Rn Rm is differentiable at a, and g : Rm Rp is → → differentiable at f(a), then the composition g f : Rn Rp is differentiable ◦ → at a, and D(g f)(a)= Dg(f(a)) Df(a) (2.21) ◦ ◦ Proof. Put b = f(a), λ = f ′(a), µ = g′(b), and u(h)= f(a + h) f(a) λ(h) (2.22) − − v(k)= g(b + k) g(b) µ(k) (2.23) − − for all h Rn and k Rm. Then we have ∈ ∈ u(h) = ǫ(h) h (2.24) | | | | v(k) = η(k) k (2.25) | | | | 14 where lim ǫ(h) = 0 (2.26) h→0 lim η(k) = 0 (2.27) k→0 Given h, we can put k such that k = f(a + h) f(a). Then we have − k = λ(h)+ u(h) [ λ + ǫ(h)] h (2.28) | | | | ≤ k k | | Thus, g f(a + h) g f(a) µ(λ(h)) = g(b + k) g(b) µ(λ(h)) (2.29) ◦ − ◦ − − − = µ(k λ(h)) + v(k) (2.30) − = µ(u(h)) + v(k) (2.31) Thus g f(a + h) g f(a) µ(λ(h)) | ◦ − ◦ − | µ ǫ(h) + [ λ + ǫ(h)]η(h) (2.32) h ≤ k k k k | | which equals 0 according to Equation 2.26 and 2.27. Exercise 1. (Spivak 2-8) Let f : R R2. Prove that f is differentiable at → a R if and only if f 1 and f 2 are, and that in this case ∈ (f 1)′(a) f ′(a)= (2.33) Ã (f 2)′(a) ! Corollary 2.2.4. If f : Rn Rm, then f is differentiable at a Rn if and → ∈ 15 only if each f i is, and (f 1)′(a) (f 2)′(a) ′ . λ (a)= . (2.34) .
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