Harmonic Morphisms - Basics

Transcription

1 Department of Mathematics Faculty of Science Lund University March 11, 2014

2 Outline Harmonic Maps in Gaussian Geometry 1 Harmonic Maps in Gaussian Geometry Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3

3 Outline Harmonic Maps in Gaussian Geometry 1 Harmonic Maps in Gaussian Geometry Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 2 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms

4 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Let W be an open subset of the complex plane C. A function φ = u + iv : W C is said to be holomorphic if it satisfies the Cauchy-Riemann equations u x = v y and u y = v x.

5 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Let W be an open subset of the complex plane C. A function φ = u + iv : W C is said to be holomorphic if it satisfies the Cauchy-Riemann equations u x = v y and u y = v x. As a direct consequence of these equations we see that 2 u x = 2 v 2 x y = 2 u and 2 v y 2 x = 2 u 2 x y = 2 v y. 2 This implies that the functions u, v : W R are harmonic, hence φ = (u + iv) = u + i v = 0.

6 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Theorem 1.1 Let u : W R be a real-valued C 2 -function defined on an open simply connected subset W of C. Then the following are equivalent, i) u is the real part of a holomorphic φ = u + iv : W C, ii) u is harmonic i.e. u = 0.

7 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Theorem 1.1 Let u : W R be a real-valued C 2 -function defined on an open simply connected subset W of C. Then the following are equivalent, i) u is the real part of a holomorphic φ = u + iv : W C, ii) u is harmonic i.e. u = 0. For the gradients u = ( u/ x, u/ y), v = ( v/ x, v/ y) we have and u, v = u v x x + u v y y = 0, u 2 = ( u x )2 + ( u y )2 = ( v x )2 + ( v y )2 = v 2.

8 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Theorem 1.1 Let u : W R be a real-valued C 2 -function defined on an open simply connected subset W of C. Then the following are equivalent, i) u is the real part of a holomorphic φ = u + iv : W C, ii) u is harmonic i.e. u = 0. For the gradients u = ( u/ x, u/ y), v = ( v/ x, v/ y) we have and u, v = u v x x + u v y y = 0, u 2 = ( u x )2 + ( u y )2 = ( v x )2 + ( v y )2 = v 2. This means that the two gradients are orthogonal and of the same length at each point in W i.e. the map φ : W C is conformal, or equivalently, φ, φ = (u + iv), (u + iv) = 0.

9 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Definition 1.2 (minimal surface) Let Σ t with t ( ɛ, ɛ) be a family of surfaces with a common boundary curve i.e. Σ 0 = Σ t for all t ( ɛ, ɛ). Then Σ 0 is said to be a minimal surface if it is a critical point for the area functional i.e. d dt ( Σ t da) t=0 = 0.

10 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Definition 1.2 (minimal surface) Let Σ t with t ( ɛ, ɛ) be a family of surfaces with a common boundary curve i.e. Σ 0 = Σ t for all t ( ɛ, ɛ). Then Σ 0 is said to be a minimal surface if it is a critical point for the area functional i.e. d dt ( Σ t da) t=0 = 0. Theorem 1.3 (zero mean curvature) A surface Σ in R 3 is minimal if and only if its mean curvature vanishes i.e. H = k1 + k2 2 0.

11 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 There is an interesting connection between the theory of minimal surfaces, harmonic functions and hence complex analysis. Theorem 1.4 (the Weierstrass representation ) Every minimal surface Σ in R 3 can locally be parametrized by a conformal and harmonic map φ : W C R 3 of the form φ : z Re z [ ] f(w) 1 g 2 (w), i(1 + g 2 (w)), 2g(w) dw, z 0 where f, g : W C C are two holomorphic functions.

12 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Definition 1.5 (harmonic morphism) The map φ = u + iv : U R 3 C is said to be a harmonic morphism if the composition f φ with any holomorphic function f : W C C is harmonic.

13 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Definition 1.5 (harmonic morphism) The map φ = u + iv : U R 3 C is said to be a harmonic morphism if the composition f φ with any holomorphic function f : W C C is harmonic. Theorem 1.6 (Jacobi 1848) The map φ = u + iv : U R 3 C is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal i.e. u = v = 0, u, v = 0 and u 2 = v 2.

14 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Definition 1.5 (harmonic morphism) The map φ = u + iv : U R 3 C is said to be a harmonic morphism if the composition f φ with any holomorphic function f : W C C is harmonic. Theorem 1.6 (Jacobi 1848) The map φ = u + iv : U R 3 C is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal i.e. u = v = 0, u, v = 0 and u 2 = v 2. Proof. (f φ) = f z φ + 2 f φ, φ = 0. z2

15 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Theorem 1.7 (the Jacobi representation ) Let f, g : W C C be holomorphic functions, then every local solution z : U R 3 C to the equation [ ] f(z(x)) 1 g 2 (z(x)), i(1 + g 2 (z(x))), 2g(z(x)), x = 1 is a harmonic morphism.

16 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Theorem 1.7 (the Jacobi representation ) Let f, g : W C C be holomorphic functions, then every local solution z : U R 3 C to the equation [ ] f(z(x)) 1 g 2 (z(x)), i(1 + g 2 (z(x))), 2g(z(x)), x = 1 is a harmonic morphism. Theorem 1.8 (Baird, Wood 1988) Every local harmonic morphism z : U C in the Euclidean R 3 is obtained this way.

17 Holomorphic Functions in One Variable Minimal Surfaces in R 3 Harmonic Morphisms in R 3 Example 1.9 (the outer disc example) Let r R + and choose g(z) = z, f(z) = 1/2irz then we obtain with the two solutions (x 1 ix 2)z 2 2(x 3 + ir)z (x 1 + ix 2) = 0 z ± r = (x3 + ir) ± x x2 2 + x2 3 r2 + 2irx 3 x 1 ix 2.

18 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Definition 2.1 (harmonic) A map φ : (M, g) (N, h) between Riemannian manifolds is said to be harmonic if it is a critial point for the energy functional E(φ) = 1 dφ 2 dvol M. 2 M

19 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Definition 2.1 (harmonic) A map φ : (M, g) (N, h) between Riemannian manifolds is said to be harmonic if it is a critial point for the energy functional E(φ) = 1 dφ 2 dvol M. 2 M The Euler-Lagrange equation for harmonic maps can be expressed, in local coordinates x on M and y on N, as the following non-linear system of PDEs τ(φ) = m i,j=1 g ij 2 φ γ x i x j m k=1 ˆΓ k φ γ ij + x k n α,β=1 Γ γ αβ φ φα φ β = 0 x i x j where φ α = y α φ and ˆΓ and Γ the Christoffel symbols on M and N, respectively.

20 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Definition 2.1 (harmonic) A map φ : (M, g) (N, h) between Riemannian manifolds is said to be harmonic if it is a critial point for the energy functional E(φ) = 1 dφ 2 dvol M. 2 M The Euler-Lagrange equation for harmonic maps can be expressed, in local coordinates x on M and y on N, as the following non-linear system of PDEs τ(φ) = m i,j=1 g ij 2 φ γ x i x j m k=1 ˆΓ k φ γ ij + x k n α,β=1 Γ γ αβ φ φα φ β = 0 x i x j where φ α = y α φ and ˆΓ and Γ the Christoffel symbols on M and N, respectively. Example 2.2 (geodesic) A curve γ : I (N, h) is harmonic if and only if it is a geodesic.

21 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Theorem 2.3 (Eells, Sampson 1964) A conformal immersion φ : (M 2, g) (N, h) of a surface is harmonic if and only if the image φ(m) is an immersed minimal surface in N.

22 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Theorem 2.3 (Eells, Sampson 1964) A conformal immersion φ : (M 2, g) (N, h) of a surface is harmonic if and only if the image φ(m) is an immersed minimal surface in N. Definition 2.4 (constant mean curvature) A surface Σ in R 3 is said to be of constant mean curvature if there exists a constant c R such that the mean curvature H of Σ satisfies H = k1 + k2 2 c.

23 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Theorem 2.3 (Eells, Sampson 1964) A conformal immersion φ : (M 2, g) (N, h) of a surface is harmonic if and only if the image φ(m) is an immersed minimal surface in N. Definition 2.4 (constant mean curvature) A surface Σ in R 3 is said to be of constant mean curvature if there exists a constant c R such that the mean curvature H of Σ satisfies H = k1 + k2 2 c. Theorem 2.5 (Ruh, Vilms 1970) A surface Σ in R 3 is of constant mean curvature if and only if its Gauss map N : Σ S 2 is harmonic.

24 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Let φ = u + iv : U C be a holomorphic function defined on an open subset U of C m. Then it satisfies the Cauchy-Riemann equations i.e. if for each k = 1, 2,..., m u = v and u = v. x k y k y k x k As a direct consequence of these equations we get 2 u = 2 v = 2 u x 2 k x k y k yk 2 and 2 v x 2 k = 2 u = 2 v. x k y k y 2 k

25 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Let φ = u + iv : U C be a holomorphic function defined on an open subset U of C m. Then it satisfies the Cauchy-Riemann equations i.e. if for each k = 1, 2,..., m u = v and u = v. x k y k y k x k As a direct consequence of these equations we get 2 u = 2 v = 2 u x 2 k x k y k yk 2 and 2 v x 2 k = 2 u = 2 v. x k y k y 2 k This implies that the functions u, v : U R are harmonic, hence φ = (u + iv) = u + i v = 0. For the two gradients u, v we have the following φ, φ = (u + iv), (u + iv) = 0. If ψ : U C is another holomorphic function then the polar identity gives 4 φ, ψ = (φ + ψ), (φ + ψ) (φ ψ), (φ ψ) φ, ψ = 0.

26 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Definition 2.6 (harmonic morphism) A map φ : (M, g) (N, h) between Riemannian manifolds is said to be a harmonic morphism if, for any harmonic function f : U R defined on an open subset U of N with φ 1 (U) non-empty, f φ : φ 1 (U) R is a harmonic function.

27 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Definition 2.6 (harmonic morphism) A map φ : (M, g) (N, h) between Riemannian manifolds is said to be a harmonic morphism if, for any harmonic function f : U R defined on an open subset U of N with φ 1 (U) non-empty, f φ : φ 1 (U) R is a harmonic function. Theorem 2.7 (Fuglede 1978) Let φ : (M, g) (N n, h) be a non-constant harmonic morphism. Then m n and M = {x M rank dφ x = n} is an open and dense subset of M.

28 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Definition 2.6 (harmonic morphism) A map φ : (M, g) (N, h) between Riemannian manifolds is said to be a harmonic morphism if, for any harmonic function f : U R defined on an open subset U of N with φ 1 (U) non-empty, f φ : φ 1 (U) R is a harmonic function. Theorem 2.7 (Fuglede 1978) Let φ : (M, g) (N n, h) be a non-constant harmonic morphism. Then m n and M = {x M rank dφ x = n} is an open and dense subset of M. Theorem 2.8 (Fuglede 1978) Let φ : (M, g) (N, h) be a non-constant harmonic morphism. If M is compact, then N is compact.

29 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Theorem 2.9 (Fuglede 1978, Ishihara 1979) A map φ : (M, g) (N, h) between Riemannian manifolds is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal.

30 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Theorem 2.9 (Fuglede 1978, Ishihara 1979) A map φ : (M, g) (N, h) between Riemannian manifolds is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal. Definition 2.10 (horizontally (weakly) conformal) A map φ : (M, g) (N, h) between Riemannian manifolds is said to be horizontally (weakly) conformal at x M if either dφ x = 0, or dφ x maps the horizontal space H x = (Ker(dφ x)) conformally onto T φ(x) N i.e. there exist a function λ : M R + 0 such that for all X, Y Hx we have h(dφ x(x), dφ x(y )) = λ 2 (x)g(x, Y ).

31 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Example 2.11 (Fuglede 1978) Let φ = u + iv : U C be a holomorphic function defined on an open subset U of C m. Then we know that φ is harmonic i.e. Furthermore φ = (u + iv) = u + i v = 0. φ, φ = (u + iv), (u + iv) = 0. This means that the two gradients are orthogonal and of the same length at each point i.e. the map φ : U C is horizontally conformal, hence a harmonic morphism.

32 Minimal Surfaces Holomorphic Functions in Several Variables Harmonic Morphisms Example 2.11 (Fuglede 1978) Let φ = u + iv : U C be a holomorphic function defined on an open subset U of C m. Then we know that φ is harmonic i.e. Furthermore φ = (u + iv) = u + i v = 0. φ, φ = (u + iv), (u + iv) = 0. This means that the two gradients are orthogonal and of the same length at each point i.e. the map φ : U C is horizontally conformal, hence a harmonic morphism. Theorem 2.12 (Baird-Eells 1981) Let φ : (M, g, J) C be a holomorphic function from a Kähler manifold. Then φ is a harmonic morphism.