Biharmonic tori in Euclidean spheres

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1 Biharmonic tori in Euclidean spheres Cezar Oniciuc Alexandru Ioan Cuza University of Iaşi Geometry Day 2017 University of Patras June 10 Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

2 Introducing the biharmonic immersions Harmonic and biharmonic maps Let (M m,g) and (N n,h) be two Riemannian manifolds C (M,N) the set of all smooth maps between M and N Energy functional E (ϕ) = E 1 (ϕ) = 1 dϕ 2 v g 2 M Euler-Lagrange equation Bienergy functional E 2 (ϕ) = 1 τ(ϕ) 2 v g 2 M Euler-Lagrange equation τ(ϕ) = τ 1 (ϕ) = trace g dϕ = 0 Critical points of E: harmonic maps τ 2 (ϕ) = ϕ τ(ϕ) trace g R N (dϕ,τ(ϕ))dϕ = 0 Critical points of E 2 : biharmonic maps Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

3 Introducing the biharmonic immersions The biharmonic equation (G.Y. 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 and R N (X,Y)Z = N X N Y Z N Y N X Z N [X,Y] Z. is a fourth-order non-linear elliptic equation; any harmonic map is biharmonic; a non-harmonic biharmonic map is called proper biharmonic; a submanifold ϕ : M N of a Riemannian manifold N is called biharmonic if the map ϕ : M N is biharmonic (ϕ is a harmonic map if and only if M is minimal). Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

4 Introducing the biharmonic immersions Biharmonic maps; non-existence results If M compact and Riem N 0 then biharmonic = harmonic (G.Y. Jiang ). A biharmonic Riemannian immersion is minimal (harmonic) if τ(ϕ) 2 is constant, i.e. it is CMC, and Riem N 0 (C.O ). study biharmonic maps into spheres. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

5 CMC biharmonic immersions in Euclidean sphere The biharmonic equation in spheres Theorem (B.Y. Chen ; C.O ) A submanifold M m in a Euclidean sphere S n of radius 1 is biharmonic if and only if H + traceb(a H ( ), ) mh = 0 4traceA ( ) H ( ) + mgrad( H 2 ) = 0, where is the Laplacian in the normal bundle, B is the second fundamental form of the submanifold and H is the mean curvature vector field. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

6 CMC biharmonic immersions in Euclidean sphere Main examples in spheres The extrinsec product ( ) ( 1 2 S m 2 1 S m 1 is proper biharmonic (G.Y. Jiang). 2 ) S m 1+m 2 +1, m 1 m 2 45 th -parallel: S m ( 1 2 ) S m+1 is proper biharmonic. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

7 CMC biharmonic immersions in Euclidean sphere Main examples in spheres Theorem (R. Caddeo, S. Montaldo, C.O ) Let ψ : M m S n (a) be a minimal submanifold, a (0,1). Then ϕ = i ψ : M S n+1 is proper biharmonic if and only if a = 1 2, where i : S n (a) S n+1 is the canonical inclusion. Theorem (R. Caddeo, S. Montaldo, C.O ) Let ψ : M m S n ( 1 2 ) be a submanifold. Then ϕ = i ψ : M S n+1 is proper biharmonic if and only if ψ is minimal. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

8 CMC biharmonic immersions in Euclidean sphere Main examples in spheres Theorem (R. Caddeo, S. Montaldo, C.O ) Let n 1, n 2 be two positive ( integers ) such that n 1 + n 2 = n 1, and let M 1 be a submanifold in S n 1 1 of dimension m 2 1, with 0 m 1 n 1, and let ( ) M 2 be a submanifold in S n of dimension m 2, with 0 m 2 n 2. Then M 1 M 2 is proper biharmonic in S n if and only if m 1 m 2 or τ(ψ 1 ) > 0 τ 2 (ψ 1 ) + 2(m 2 m 1 )τ(ψ 1 ) = 0 τ 2 (ψ 2 ) 2(m 2 m 1 )τ(ψ 2 ) = 0 τ(ψ 1 ) = τ(ψ 2 ) = constant, ( ) ( ) where ψ 1 : M 1 S n 1 1 and ψ 2 2 : M 2 S n are the Riemannian immersions. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

9 CMC biharmonic immersions in Euclidean sphere Main examples in spheres Corollary If τ(ψ 1 ) = τ(ψ 2 ) = 0, then M m 1 only if m 1 m 2. 1 Mm 2 2 is proper biharmonic in S n if and Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

10 CMC biharmonic immersions in Euclidean sphere Constraint on H Theorem (C.O ) Let ϕ : M m S n be a CMC proper biharmonic immersion. Then (i) H (0,1]; (ii) H = 1 if and only if ϕ induces a minimal immersion ψ of M m in the small hypersphere S n 1 ( 1 2 ) S n. Interesting case: H (0, 1). Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

11 CMC biharmonic immersions in Euclidean sphere Type decomposition (B.Y. Chen) A Riemannian immersion φ : M m R n+1 is called of finite type if it can be expressed as a finite sum of R n+1 -valued eigenmaps of the Laplacian of (M,g), i.e. where φ 0 R n+1 is a constant vector and φ = φ 0 + φ t1 + + φ tk, (1) φ ti = λ ti φ ti, i = 1,...,k. If, in particular, all eigenvalues λ ti are mutually distinct, the submanifold is said to be of k-type and (1) is called the spectral decomposition of φ. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

12 CMC biharmonic immersions in Euclidean sphere Types for biharmonic Theorem (A. Balmuş, S. Montaldo, C.O ; E. Loubeau, C.O ) Let ϕ : M m S n be a proper biharmonic immersion. Denote by φ = i ϕ : M R n+1 the immersion of M in R n+1, where i : S n R n+1 is the canonical inclusion map. Then (i) The immersion φ is of 1-type if and only if H = 1. In this case, φ = φ 0 + φ t1, with φ t1 = 2mφ t1, φ 0 is a constant vector. Moreover, ( ) φ 0,φ t1 = 0 at any point, φ 0 = φ t1 = 1 2 and ψ t1 : M S n is a minimal immersion. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

13 CMC biharmonic immersions in Euclidean sphere Types for biharmonic Theorem (Continued) (ii) The immersion φ is of 2-type if and only if H is constant, H (0,1). In this case φ = φ t1 + φ t2, with φ t1 = m(1 H )φ t1, φ t2 = m(1 + H )φ t2 and φ t1 = 1 2 φ H H, φ t 2 = 1 2 φ 1 2 H H. Moreover, φ t1,φ t2 = 0, φ t1 = φ t2 = 1 2 and ψ ti : (M,g) S n ( 1 2 ), i = 1,2, are harmonic maps with constant density energy. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

14 CMC biharmonic immersions in Euclidean sphere Type decomposition uniqueness Theorem (E. Loubeau, C.O ) Let ϕ 1,ϕ 2 : M m S n be two CMC proper biharmonic immersions. If ϕ 1 and ϕ 2 agree on an open subset of M, then they agree everywhere. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

15 Biconservative submanifolds Definition of biconservative submanifolds A submanifold ϕ : M N is called biconservative if τ 2 (ϕ) = 0. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

16 CMC biharmonic surfaces in Euclidean sphere Properties of biharmonic (biconservative) surfaces in spheres Theorem (E. Loubeau, C.O ) Let ϕ : M 2 N n be a CMC proper biharmonic (biconservative) surface. If M is compact and K M 0, then A H = 0 and M is flat or pseudo-umbilical. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

17 CMC biharmonic surfaces in Euclidean sphere Properties of biharmonic (biconservative) surfaces in spheres Hopf differential Let ϕ : M 2 N n be a proper biharmonic (biconservative) surface with mean curvature vector field H. Let z be a complex coordinate on M 2. Then the function B( z, z ),H is holomorphic if and only if the norm of H is constant (CMC). Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

18 CMC biharmonic surfaces in Euclidean sphere Properties of biharmonic (biconservative) surfaces in spheres Let ϕ : M 2 N n be a CMC proper biharmonic (biconservative) surface. If M 2 is a topological sphere, then it is pseudo-umbilical. Let ϕ : M 2 N n be a CMC proper biharmonic (biconservative) surface. If M is not pseudo-umbilical, then its pseudo-umbilical points are isolated. Let ϕ : M 2 N n be a CMC proper biharmonic (biconservative) surface. Assume M is compact and has no pseudo-umbilical point, then it is topologically a torus. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

19 CMC biharmonic surfaces in Euclidean sphere Constant Gaussian curvature Boruvka spheres and their uniqueness (Bryant); Bryant results for minimal surfaces in spheres which are flat or have negative constant Gaussian curvature; Miyata results for 2-type immersions. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

20 CMC biharmonic surfaces in Euclidean sphere Positive constant curvature Theorem (E. Loubeau, C.O ) A Riemannian immersion ϕ is a CMC proper biharmonic map from a surface with constant positive Gaussian curvature in S n if and only if n is odd and it is the diagonal map where φ = φ t1 + φ t2 = (αψ 1,βψ 2 ), ψ 1 : S 2 (r) S 2n 1 (r 1 ) ψ 2 : S 2 (r) S 2n 2 (r 2 ) are Boruvka minimal immersions and the parameters α, β, r 1, r 2, r are given by Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

21 CMC biharmonic surfaces in Euclidean sphere Positive constant curvature Theorem (Continued) α 2 = q 1 and β 2 = q 2 q 1 + q 2 q 1 + q 2 q1 + q 2 q1 + q 2 r 1 = and r 2 = 2q 1 2q 2 r = 1 2 q1 + q 2, with q 1 = n 1 (n 1 + 1) and q 2 = n 2 (n 2 + 1), and n 1 n 2. Moreover, H 2 = (q 1 q 2 ) 2 (q 1 +q 2 ) 2 and ϕ is pseudo-umbilical. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

22 CMC biharmonic surfaces in Euclidean sphere Negative constant curvature There can be no CMC proper biharmonic surface in a sphere with K M constant and K M < 0. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

23 The flat case CMC biharmonic surfaces in Euclidean sphere Local description: Let D be a small disk about the origin in R 2 and ϕ : D S n a CMC proper biharmonic immersion with H (0,1). Then (i) n is odd, n 5; (ii) ϕ extends uniquely to a CMC proper biharmonic immersion of R 2 in S n ; (iii) φ = i ϕ : R 2 R n+1 Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

24 The flat case CMC biharmonic surfaces in Euclidean sphere where φ(z) = 1 m ( ) λ1 2 Rk e 2 (µ k z µ k z) λ1 Z k + e 2 ( µ k z+ µ k z) Z k k=1 + 1 m ( R λ2 j e 2 (η j z η j z) λ2 W j + e 2 ( η j z+ η j z) W j ), (2) 2 j=1 Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

25 The flat case CMC biharmonic surfaces in Euclidean sphere Z k = 1 2 (E 2k 1 ie 2k ), k = 1,...,m, i 2 = 1, W j = 1 2 ( E2(m+j) 1 ie 2(m+j) ), j = 1,...,m, {E 1,...,E 2m+2m } is an orthonormal basis of R n+1, n = 2m + 2m 1, λ 1 = 2(1 H ), λ 2 = 2(1 + H ), H constant, H (0,1), k R k = 1, j R j = 1, R k > 0, R j > 0, (1 H ) k µ 2 k R k + (1 + H ) j η 2 j R j = 0, {±µ k } m k=1 are 2m complex numbers of norm 1, {±η j } m j=1 are 2m complex numbers of norm 1. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

26 The flat case CMC biharmonic surfaces in Euclidean sphere (1 H ) k µ k 2 R k + (1 + H ) ηj 2 R j = 0. (3) j Symmetries of the solutions Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

27 The flat case Theorem CMC biharmonic surfaces in Euclidean sphere Let ϕ : R 2 S n be a CMC proper biharmonic immersion with H (0,1). Then it is a diagonal map φ = φ t1 + φ t2 = (φ 1,φ 2 ), where φ 1 : R 2 S 2m 1 ( 1 2 ) and φ 2 : R 2 S 2m 1 ( 1 2 ) are harmonic maps with constant density energy, n = 2m + 2m 1. Theorem Let ϕ : R 2 S n be a CMC proper biharmonic immersion with H (0,1). Then it is pseudo-umbilical if and only if φ 1, = 1 H 2, and φ2, = 1 + H,. 2 Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

28 The flat case CMC biharmonic surfaces in Euclidean sphere AIMS Solve completely (3). Quotient the biharmonic immersion (2) to a cylinder. Quotient the biharmonic immersion (2) to a torus. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

29 CMC flat biharmonic surfaces in S 5 n = 5; Structure Theorem Take h (0,1). Then there is a one-parameter family of CMC proper biharmonic surfaces ϕ h,ρ = ϕ ρ : R 2 S 5 with mean curvature H = h, ρ [ 0, 1 h 1 2 arccos 1+h], Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

30 CMC flat biharmonic surfaces in S 5 n = 5; Structure Theorem φ ρ = i ϕ ρ : R 2 R 6 can be written as φ ρ (z) = 1 ) λ1 (e 2 (z z) λ1 Z 1 + e 2 ( z+ z) Z R j=1( j e λ2 2 (η j z η j z) W j + R j e λ2 2 ( η jz+ η j z) W j ), where Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

31 CMC flat biharmonic surfaces in S 5 n = 5; Structure Theorem Z 1 = 1 2 (E 1 ie 2 ), W j = 1 ( ) 2 E2(1+j) 1 ie 2(1+j), j = 1,2, {E 1,...,E 6 } is an orthonormal basis of R 6, λ 1 = 2(1 h), λ 2 = 2(1 + h), R 1, R 2, η 1 = e iρ and η 2 = e i ρ are given by: Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

32 CMC flat biharmonic surfaces in S 5 n = 5; Structure Theorem if ρ ( 0, 1 h 1 2 arccos 1+h], and R 1 = 1 ( ) h 1 2 h+1 2 ( 1 + h 1 h+1 cos2ρ), ) 2 R 1 ( h 1 h+1 2 = 1 2 ( 1 + h 1 h+1 cos2ρ), ( ρ = arctan 1 ), htanρ Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

33 CMC flat biharmonic surfaces in S 5 n = 5; Structure Theorem if ρ = 0. R 1 = h 1 + h, R 2 = h, ρ = π 2, Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

34 CMC flat biharmonic surfaces in S 5 n = 5; Structure Theorem Conversely, assume that ϕ : R 2 S 5 is a CMC proper biharmonic surface with mean curvature H = h (0,1). Then, up to isometries of R 2 and R 6, φ = i ϕ : R 2 R 6 is one of the above maps. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

35 CMC flat biharmonic surfaces in S 5 Existence of biharmonic planes Let h (0,1), then there exist proper biharmonic immersions of R 2 in S 5 with CMC equal to h. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

36 CMC flat biharmonic surfaces in S 5 Existence of biharmonic cylinders Let h (0,1), then there exist proper biharmonic cylinders in S 5 with CMC equal to h. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

37 CMC flat biharmonic surfaces in S 5 Biharmonic tori in S 5 Theorem The CMC proper biharmonic immersion ϕ h,ρ : R 2 S 5, ρ to a torus if and only if either (i) ρ = 0 and h = 1 b 1 + b, [ 0, 1 2 where b = ( ) r 2, t r,t N, with r < t and (r,t) = 1; or ( ] (ii) ρ 0, 2 1 h 1 arccos 1+h is a constant depending on a and b and h = 1 (a b) (a b) 2 + 2(a + b), ] h 1 arccos 1+h, quotients where a = ( p q ) 2 and b = ( rt ) 2, with p,q,r,t N, such that 0 b a < 1. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

38 CMC flat biharmonic surfaces in S 5 Biharmonic tori in S 5 Theorem (Continued) Moreover, in this case, the corresponding lattice Λ ψh,ρ is given by Λ ψh,0 ={l r v 2 + k v 1 : k,l Z} ={l t ṽ 2 + k v 1 : k,l Z}, where and v 1 = ( π 1 + b,0) ( 1 + b ) v 2 = 0,π b ṽ 2 = (0,π 1 + b); or Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

39 CMC flat biharmonic surfaces in S 5 Biharmonic tori in S 5 Theorem (Continued) where { Λ ψh,ρ = m v 2 + n v 1 : m,n Z s.t. m q p n qr } pt Z { = m ṽ 2 + k ṽ 1 : m,k Z s.t. m t r k pt } qr Z, ( π (a b) v 1 = 2 + a + b ),0 a ( v 2 = π b(1 a + b) a((a b) 2 + a + b), π 1 + (a b) 2 + 2(a + b) ) (a b) 2 + a + b ( π (a b) ṽ 1 = 2 + a + b ),0 b ( ṽ 2 = π a(1 + a b) b((a b) 2 + a + b), π 1 + (a b) 2 + 2(a + b) (a b) 2 + a + b ). Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

40 CMC flat biharmonic surfaces in S 5 Biharmonic tori in S 5 Theorem Let h (0,1). Then there exists a CMC proper biharmonic immersion from some torus T 2 in S 5 with mean curvature h if and only if either (i) h = 1 b 1 + b, (ii) where b = ( r t ) 2, r,t N, with r < t; or h = 1 (a b) (a b) 2 + 2(a + b), where a = ( p q) 2 and b = ( r t ) 2, with p,q,r,t N, such that 0 b a < 1. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

41 CMC flat biharmonic surfaces in S n Flat tori biharmonically immersed in S n AIM Given a flat torus T 2, does it admit a biharmonic immersion in some S n? Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

42 CMC flat biharmonic surfaces in S n A class of CMC biharmonic rectangular tori Theorem (D. Fetcu, E. Loubeau, C.O ) Consider a rectangular lattice Λ = {(2πk,2πlθ) : k,l Z} with a side of length 2π and the other of length 2πθ 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) if and only if θ 2 = q2 1 + q2 2 2 where q 1,q 2 N and q 1 < q 2. In this case 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. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

43 CMC flat biharmonic surfaces in S n A class of CMC biharmonic rectangular tori Theorem (Continued) Moreover, the CMC proper biharmonic immersion from T 2 in S n corresponds to the map φ : R 2 R n+1 determined by one of the following sets of data: when n = 5 R 1 = 1, R 1 = 1 2 ( 1 ) ( 1 2h, R h = 1 ) 1 2h h 1 2h µ 1 = 2(1 h) + i 1 + 2h, η 1 = 2(1 h) 2(1 + h) + i, η 2 = η 1, 2(1 + h) where h (0, 1 2 ]; Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

44 CMC flat biharmonic surfaces in S n A class of CMC biharmonic rectangular tori Theorem (Continued) when n = 7 R 1,2 = 1 ( ) ω 1 ±, R 2 1 2h 1,2 = 1 ( ω 1 ), h 1 2h µ 1 = 2(1 h) + i 1 + 2h, µ 2 = µ 1, η 1 = 2(1 h) 2(1 + h) + i,η 2 = η 1, 2(1 + h) where ω ( 1 2h, 1 2h) and h (0, 1 2 ). Remark The proof relies on explicitly finding all admissible CMC proper biharmonic immersions of T 2 in S n. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

45 CMC flat biharmonic surfaces 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. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

46 CMC flat biharmonic surfaces in S n CMC biharmonic square tori in S n Theorem Consider a square lattice Λ = {(2πka,2πla) : k,l Z} with the side of length 2πa and the torus T 2 = R 2 /Λ, where a R +. Then we have (i) T 2 admits a proper biharmonic immersion in S n, n 3 (mod 4), with constant mean curvature h (0,1) if and only if 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. (ii) 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 = q2 p 2 p 2 +q 2 for any odd n, 5 n r 2 (p 2 ) + r 2 (q 2 ) 1. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

47 CMC flat biharmonic surfaces in S n CMC biharmonic square tori in S n Remark 3 One obtains that a 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. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

48 CMC flat biharmonic surfaces in S n CMC biharmonic square 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 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. Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49

49 CMC flat biharmonic surfaces in S n Thank You! Cezar Oniciuc (UAIC) Biharmonic tori in Euclidean spheres Patras, June 10, / 49