Constant mean curvature biharmonic surfaces

Transcription

1 Constant mean curvature biharmonic surfaces Dorel Fetcu Gheorghe Asachi Technical University of Iaşi, Romania Brest, France, May 2017 Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

2 The harmonic and biharmonic problems Harmonic and biharmonic maps Let ϕ : (M,g) (N,h) be a smooth map. Energy functional E (ϕ) = E 1 (ϕ) = 1 dϕ 2 v g 2 M Euler-Lagrange equation τ(ϕ) = τ 1 (ϕ) = trace g dϕ = 0 Critical points of E: harmonic maps Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

3 The harmonic and biharmonic problems Harmonic and biharmonic maps Let ϕ : (M,g) (N,h) be a smooth map. Energy functional E (ϕ) = E 1 (ϕ) = 1 dϕ 2 v g 2 M Euler-Lagrange equation τ(ϕ) = τ 1 (ϕ) = trace g dϕ = 0 Bienergy functional E 2 (ϕ) = 1 τ(ϕ) 2 v g 2 Euler-Lagrange equation τ 2 (ϕ) = ϕ τ(ϕ) trace g R N (dϕ,τ(ϕ))dϕ = 0 M Critical points of E: harmonic maps Critical points of E 2 : biharmonic maps Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

4 The harmonic and biharmonic problems The biharmonic equation The biharmonic equation (Jiang ) τ 2 (ϕ) = ϕ τ(ϕ) trace g R N (dϕ,τ(ϕ))dϕ = 0 where ϕ ( = trace g ϕ ϕ ϕ ) is the rough Laplacian on sections of ϕ 1 TN Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

5 The harmonic and biharmonic problems The biharmonic equation The biharmonic equation (Jiang ) τ 2 (ϕ) = ϕ τ(ϕ) trace g R N (dϕ,τ(ϕ))dϕ = 0 where ϕ = trace g ( ϕ ϕ ϕ ) is the rough Laplacian on sections of ϕ 1 TN it is a fourth-order non-linear elliptic equation any harmonic map is biharmonic a non-harmonic biharmonic map is proper-biharmonic a submanifold i : M N is a biharmonic submanifolds if the immersion i is biharmonic Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

6 The harmonic and biharmonic problems Biharmonic submanifolds The biharmonic equation Theorem (Balmuş, Montaldo, Oniciuc ) A submanifold Σ m in a Riemannian manifold N is biharmonic iff { H + traceσ(,a H ) + trace(r N (,H) ) = 0 m 2 grad H 2 + 2traceA H( ) + 2trace(R N (,H) ) = 0, where is the Laplacian in the normal bundle. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

7 Biharmonic tori in spheres CMC biharmonic immersions in S n Theorem (Loubeau, Oniciuc ) Let D be a small disk about the origin in the Euclidean plane R 2 and φ : D S n be a CMC proper-biharmonic immersion with mean curvature h = H (0,1). Then n is odd, n 5, and φ extends uniquely to a CMC proper-biharmonic immersion of R 2 into S n. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

8 Biharmonic tori in spheres CMC biharmonic immersions in S n Theorem (Loubeau, Oniciuc ) Let D be a small disk about the origin in the Euclidean plane R 2 and φ : D S n be a CMC proper-biharmonic immersion with mean curvature h = H (0,1). Then n is odd, n 5, and φ extends uniquely to a CMC proper-biharmonic immersion of R 2 into S n. Remark The above result also gives the explicit expression of ψ = i φ : R 2 R n Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

9 The structure theorem in S 5 Biharmonic tori in spheres The structure theorem in S 5 Theorem (Loubeau, Oniciuc ) For a given h (0,1) there is unique one-parameter family of CMC proper-biharmonic surfaces φ h,ρ = φ ρ : R 2 S 5 with mean curvature h and ρ [0,(1/2)arccos((h 1)/(1 + h))]. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

10 The structure theorem in S 5 Biharmonic tori in spheres The structure theorem in S 5 Theorem (Loubeau, Oniciuc ) The CMC proper-biharmonic immersion φ h,ρ : R 2 S 5 quotients to a torus if and only if either (a) ρ = 0 and h = 1 b 1 + b, where b = r 2 /t 2, r,t N, with r < t and (r,t) = 1; or (b) ρ (0,(1/2)arccos((h 1)/(1 + h))] is a constant depending on a and b and h = 1 (a b) (a b) 2 + 2(a + b), where a = p 2 /q 2 and b = r 2 /t 2, with p,q,r,t N, such that 0 b a < 1. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

11 The structure theorem in S 5 Biharmonic tori in spheres The structure theorem in S 5 Theorem (Loubeau, Oniciuc ) The CMC proper-biharmonic immersion φ h,ρ : R 2 S 5 quotients to a torus if and only if either (a) ρ = 0 and h = 1 b 1 + b, where b = r 2 /t 2, r,t N, with r < t and (r,t) = 1; or (b) ρ (0,(1/2)arccos((h 1)/(1 + h))] is a constant depending on a and b and h = 1 (a b) (a b) 2 + 2(a + b), where a = p 2 /q 2 and b = r 2 /t 2, with p,q,r,t N, such that 0 b a < 1. Remark The corresponding lattices Λ φh,0 and Λ φh,ρ were also explicitly determined. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

12 Biharmonic tori in spheres Biharmonic immersions of tori in S n A class of CMC biharmonic rectangular tori Theorem (F., Loubeau, Oniciuc ) Consider a rectangular lattice Λ = {(2πk,2πlθ) : k,l Z} and the torus T 2 = R 2 /Λ, where θ R +. Then T 2 admits a proper-biharmonic immersion in S n with constant mean curvature h (0,1) iff where q 1,q 2 N and q 1 < q 2. In this case θ 2 = (q q2 2 )/2 and n {5,7}, h = q2 2 q2 1 2(q q2 2 ), with q 1 0 when n = 5 and q 1 > 0 when n = 7. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

13 Biharmonic tori in spheres Biharmonic immersions of tori in S n A class of CMC biharmonic rectangular tori Theorem (F., Loubeau, Oniciuc ) Consider a rectangular lattice Λ = {(2πk,2πlθ) : k,l Z} and the torus T 2 = R 2 /Λ, where θ R +. Then T 2 admits a proper-biharmonic immersion in S n with constant mean curvature h (0,1) iff where q 1,q 2 N and q 1 < q 2. In this case θ 2 = (q q2 2 )/2 and n {5,7}, h = q2 2 q2 1 2(q q2 2 ), with q 1 0 when n = 5 and q 1 > 0 when n = 7. Remark The proof relies on explicitly finding all admissible CMC proper-biharmonic immersions of T 2 in S n. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

14 Biharmonic tori in spheres Biharmonic immersions of tori in S n A class of CMC biharmonic rectangular tori Remark For 0 < q 2 1 < q2 2 the same torus can be immersed in S5 and S 7 as a CMC proper-biharmonic surface, with the same constant mean curvature. Remark The same rectangular torus can be immersed in S 5 (or S 7 ) as a CMC proper-biharmonic surface in different ways with different mean curvatures. Remark A rectangular torus with both sides of length less than 1/ 2 cannot be immersed in a sphere S n as a CMC proper-biharmonic surface. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

15 Biharmonic tori in spheres CMC biharmonic square tori in S n Biharmonic immersions of tori in S n Theorem (F., Loubeau, Oniciuc ) Consider a square lattice Λ = {(2πka,2πla) : k,l Z} and the torus T 2 = R 2 /Λ, where a R +. Then we have (a) T 2 admits a proper-biharmonic immersion in S n, n 3 (mod 4), with constant mean curvature h (0,1) iff and 4a 2 = p q2 1 + p2 2 + q2 2, h = p2 2 + q2 2 p2 1 q2 1 p q2 1 + p2 2 +, q2 2 7 n r 2 (p q2 1 ) + r 2(p q2 2 ) 1, where r 2 (p) is the number of representations of p N as the sum of two squares of integers and p 1,q 1,p 2,q 2 N such that 0 < p q2 1 < p2 2 + q2 2. (b) If 4a 2 = p 2 + q 2, where p,q N such that 0 < p < q, then T 2 admits a CMC proper-biharmonic immersion in S n with h = (q 2 p 2 )/(p 2 + q 2 ) for any odd n, 5 n r 2 (p 2 ) + r 2 (q 2 ) 1. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

16 Biharmonic tori in spheres Biharmonic immersions of tori in S n CMC biharmonic square tori in S n Remark One obtains that a 3/2. Moreover, if a = 3/2, the corresponding square torus can be immersed only in S 7, in a unique manner. Remark While any positive integer can be written as a sum of four squares (not necessarily satisfying the condition in the theorem), a positive integer can be written as a sum of two squares if and only if each of its prime factors of the form 4p 1 occurs with an even power in its prime factorization. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

17 Biharmonic tori in spheres CMC biharmonic square tori in S n Biharmonic immersions of tori in S n As positive integers p and q can be chosen such that r 2 (p 2 ) + r 2 (q 2 ) is arbitrarily large, we have the following Theorem (F., Loubeau, Oniciuc ) For any sphere S n, with n odd, n 5, there exists a square torus that can be immersed in S n as a CMC proper-biharmonic surface. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

18 PMC biharmonic surfaces in CP n Surfaces with parallel mean curvature Let Σ be a surface of a Riemannian manifold N N X Y = X Y + σ(x,y) (Eq. Gauss) N X V = A V X + X V (Eq. Weingarten) Definition If the mean curvature vector field H = 1 2 traceσ is parallel in the normal bundle, i.e., H = 0, then Σ is called a PMC surface. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

19 PMC biharmonic surfaces in CP n Curves in complex space forms A curve γ : I R N n (c) parametrized by arc-length is called a Frenet curve of osculating order r, 1 r 2n, if there exist r orthonormal vector fields {E 1 = γ,...,e r } along γ such that N E 1 E 1 = κ 1 E 2, N E 1 E i = κ i 1 E i 1 +κ i E i+1,..., N E 1 E r = κ r 1 E r 1 where the functions κ i > 0 are the curvatures of γ κ i = constant > 0: helix of order r r = 2: circle r = 3: helix the complex torsions of γ are τ ij = E i,je j, 1 i < j r a helix of order r is a holomorphic helix if τ ij = constant Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

20 PMC biharmonic surfaces in CP n PMC surfaces in complex space forms Let Σ be a PMC surface in a complex space form N n (c) (F ) The (2,0)-part of the quadratic form Q defined on Σ by Q(X,Y) = 8 H 2 A H X,Y + 3c X,T Y,T, where T is the tangent part of JH, is holomorphic consider ( ) 3c S = 8 H 2 A H + 3c T, T 2 T H 4 Id then SX,Y = Q(X,Y) traceq X,Y 2 and (F., Pinheiro ) 1 2 S 2 = 2K S 2 + S 2 Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

21 PMC biharmonic surfaces in CP n PMC surfaces in complex space forms Theorem (F., Pinheiro ) Let Σ be a complete non-minimal PMC surface with K 0 in N n (c), c 0. Then one of the following holds: 1 the surface is flat; 2 there exists a point p Σ such that K(p) > 0 and Q (2,0) vanishes. Remark For a surface Σ as in the above theorem we have S = 0. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

22 PMC biharmonic surfaces in CP n The general conditions for biharmonicity Theorem (Balmuş, Montaldo, Oniciuc ) A submanifold Σ m in a Riemannian manifold N is biharmonic iff { H + traceσ(,a H ) + trace(r N (,H) ) = 0 m 2 grad H 2 + 2traceA H( ) + 2trace(R N (,H) ) = 0, where is the Laplacian in the normal bundle. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

23 PMC biharmonic surfaces in CP n PMC biharmonic surfaces Proposition Let Σ be a PMC surface in a complex space form N n (c). Then Σ is biharmonic iff traceσ(,a H ) = ρ ( 2H 3(JT) ) and (JT) = 0 4 where T is the tangent part of JH. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

24 PMC biharmonic surfaces in CP n PMC biharmonic surfaces Proposition Let Σ be a PMC surface in a complex space form N n (c). Then Σ is biharmonic iff traceσ(,a H ) = ρ ( 2H 3(JT) ) and (JT) = 0 4 where T is the tangent part of JH. Remark If Σ is a PMC proper-biharmonic surface in N n (c), then N n (c) = CP n (c), since 0 < A H 2 = c ( 2 H T 2) 4 Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

25 PMC biharmonic surfaces in CP n PMC biharmonic surfaces Proposition (F., Pinheiro ) Let Σ be a complete PMC proper-biharmonic surface in CP n (c). If T 0, then Σ is totally real and T = 0. Moreover, if K 0, then K = 0 and A H = 0. If T = 0 and K 0, then n 3 and Σ is pseudo-umbilical and totally real. Moreover, H = c/2. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

26 PMC biharmonic surfaces in CP n The classification theorem Theorem (F., Pinheiro ) Let Σ be a complete PMC proper-biharmonic surface with K 0 in CP n (c). Then Σ is totally real and either 1 Σ is pseudo-umbilical and H = c/2; or 2 Σ is a complete Lagrangian PMC proper-biharmonic surface in CP 2 (c); or 3 Σ is the product between two particular curves in CP 3 (c), a holomorphic circle and a holomorphic helix of order 4. Moreover, these curves always exist and are unique up to holomorphic isometries. Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21

27 References References D. Fetcu, E. Loubeau, and C. Oniciuc, Biharmonic tori in spheres, Differential Geom. Appl., to appear. D. Fetcu and A. L. Pinheiro, Biharmonic surfaces with parallel mean curvature in complex space forms, Kyoto J. Math. 55 (2015), Dorel Fetcu (TUIASI) CMC biharmonic surfaces Brest, May / 21