Biharmonic pmc surfaces in complex space forms

Transcription

1 Biharmonic pmc surfaces in complex space forms Dorel Fetcu Gheorghe Asachi Technical University of Iaşi, Romania Varna, Bulgaria, June 016 Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

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 v g M Euler-Lagrange equation τ(ϕ) = τ 1 (ϕ) = trace g dϕ = 0 Critical points of E: harmonic maps Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June 016 / 30

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 v g M Euler-Lagrange equation τ(ϕ) = τ 1 (ϕ) = trace g dϕ = 0 Bienergy functional E (ϕ) = 1 τ(ϕ) v g Euler-Lagrange equation τ (ϕ) = ϕ τ(ϕ) trace g R N (dϕ,τ(ϕ))dϕ = 0 M Critical points of E: harmonic maps Critical points of E : biharmonic maps Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June 016 / 30

4 The harmonic and biharmonic problems The biharmonic equation The biharmonic equation (Jiang, 1986) τ (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) Biharmonic pmc surfaces Varna, June / 30

5 The harmonic and biharmonic problems The biharmonic equation The biharmonic equation (Jiang, 1986) τ (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 is a fourth-order non-linear elliptic equation any harmonic map is biharmonic a non-harmonic biharmonic map is called proper biharmonic the biharmonic submanifolds M of a given space N are the submanifolds such that the inclusion map i : M N is biharmonic (the inclusion map i : M N is harmonic if and only if M is minimal) Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

6 The harmonic and biharmonic problems The biharmonic equation The biharmonic equation Theorem (Balmuş-Montaldo-Oniciuc, 01) A submanifold Σ m in a Riemannian manifold N, with second fundamental form σ, mean curvature vector field H, and shape operator A, is biharmonic if and only if { H + traceσ(,a H ) + trace(r N (,H) ) = 0 m grad H + tracea H( ) + trace(r N (,H) ) = 0, where is the Laplacian in the normal bundle. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

7 The harmonic and biharmonic problems Biharmonic submanifolds in Euclidean spaces Biharmonic submanifolds in Euclidean spaces Definition (Chen) R N = 0 τ (ϕ) = ϕ τ(ϕ) A submanifold i : M R n is biharmonic if it has harmonic mean curvature vector field, i.e., i H = 0 i τ(i) = 0. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

8 The harmonic and biharmonic problems Biharmonic submanifolds in Euclidean spaces Non existence of proper biharmonic submanifolds For any of the following classes of submanifolds the biharmonicity is equivalent to minimality: submanifolds of N 3 (c), c 0 (Chen/Caddeo - Montaldo - Oniciuc) curves of N n (c), c 0 (Dimitric/Caddeo - Montaldo - Oniciuc) submanifolds of finite type in R n (Dimitric) hypersurfaces of R n with at most two principal curvatures (Dimitric) pseudo-umbilical submanifolds of N n (c), c 0, n 4 (Dimitric/Caddeo - Montaldo - Oniciuc) hypersurfaces of R 4 (Hasanis - Vlachos) spherical submanifolds of R n (Chen) Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

9 The harmonic and biharmonic problems Biharmonic submanifolds in Euclidean spaces Non existence of proper biharmonic submanifolds For any of the following classes of submanifolds the biharmonicity is equivalent to minimality: submanifolds of N 3 (c), c 0 (Chen/Caddeo - Montaldo - Oniciuc) curves of N n (c), c 0 (Dimitric/Caddeo - Montaldo - Oniciuc) submanifolds of finite type in R n (Dimitric) hypersurfaces of R n with at most two principal curvatures (Dimitric) pseudo-umbilical submanifolds of N n (c), c 0, n 4 (Dimitric/Caddeo - Montaldo - Oniciuc) hypersurfaces of R 4 (Hasanis - Vlachos) spherical submanifolds of R n (Chen) Chen s conjecture (still open) Any biharmonic submanifold of the Euclidean space is minimal. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

10 The harmonic and biharmonic problems Biharmonic submanifolds in Euclidean spaces Non existence of proper biharmonic submanifolds For any of the following classes of submanifolds the biharmonicity is equivalent to minimality: submanifolds of N 3 (c), c 0 (Chen/Caddeo - Montaldo - Oniciuc) curves of N n (c), c 0 (Dimitric/Caddeo - Montaldo - Oniciuc) submanifolds of finite type in R n (Dimitric) hypersurfaces of R n with at most two principal curvatures (Dimitric) pseudo-umbilical submanifolds of N n (c), c 0, n 4 (Dimitric/Caddeo - Montaldo - Oniciuc) hypersurfaces of R 4 (Hasanis - Vlachos) spherical submanifolds of R n (Chen) Chen s conjecture (still open) Any biharmonic submanifold of the Euclidean space is minimal. Generalized Chen s Conjecture (still open) Biharmonic submanifolds of N n (c), n > 3, c 0, are minimal. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

11 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n (Jiang, 1986/ Caddeo - Montaldo - Oniciuc, 00) The composition property S n 1 (a) biharmonic S n a = 1 Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

12 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n (Jiang, 1986/ Caddeo - Montaldo - Oniciuc, 00) The composition property S n 1 (a) biharmonic S n a = 1 S n 1 ( 1 ) i S n biharmonic Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

13 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n (Jiang, 1986/ Caddeo - Montaldo - Oniciuc, 00) The composition property S n 1 (a) biharmonic S n a = 1 M minimal S n 1 ( 1 ) i S n biharmonic Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

14 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n (Jiang, 1986/ Caddeo - Montaldo - Oniciuc, 00) The composition property S n 1 (a) biharmonic S n a = 1 M minimal S n 1 ( 1 ) biharmonic i S n biharmonic Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

15 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n (Jiang, 1986/ Caddeo - Montaldo - Oniciuc, 00) The composition property S n 1 (a) biharmonic S n a = 1 Properties M minimal biharmonic S n 1 ( 1 ) i S n biharmonic M has parallel mean curvature vector field and H = 1 M is pseudo-umbilical in S n, i.e., A H = H Id Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

16 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n The product composition property S n 1(a) S n (b) biharmonic S n a = b = 1 and n 1 n n 1 + n = n 1, a + b = 1 Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

17 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n The product composition property S n 1(a) S n (b) biharmonic S n a = b = 1 and n 1 n n 1 + n = n 1, a + b = 1 S n 1( 1 ) S n ( 1 ) n 1 + n = n 1 i S n biharmonic Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

18 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n The product composition property S n 1(a) S n (b) biharmonic S n a = b = 1 and n 1 n n 1 + n = n 1, a + b = 1 M m 1 1 Mm minimal S n 1( 1 ) S n ( 1 ) i S n biharmonic n 1 + n = n 1, m 1 m Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

19 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n The product composition property S n 1(a) S n (b) biharmonic S n a = b = 1 and n 1 n n 1 + n = n 1, a + b = 1 M m 1 1 Mm minimal S n 1( 1 ) S n ( 1 ) biharmonic i S n biharmonic n 1 + n = n 1, m 1 m Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

20 The harmonic and biharmonic problems Main examples of biharmonic submanifolds in S n Main examples of biharmonic submanifolds in S n The product composition property S n 1(a) S n (b) biharmonic S n a = b = 1 and n 1 n n 1 + n = n 1, a + b = 1 M m 1 1 Mm minimal S n 1( 1 ) S n ( 1 ) biharmonic i S n biharmonic n 1 + n = n 1, m 1 m Properties M 1 M has parallel mean curvature vector field and H (0,1) M 1 M is not pseudo-umbilical in S n Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

21 Biharmonic submanifolds of CP n Complex space forms Complex space forms Definition A complex space form is a n-dimensional Kähler manifold N n (ρ) of constant holomorphic sectional curvature ρ. A complex space form N n (ρ) is either: the complex projective space CP n (ρ), if ρ > 0 the complex Euclidean space C n, if ρ = 0 the complex hyperbolic space CH n (ρ), if ρ < 0 The curvature tensor R N (U,V)W = ρ 4 { V,W U U,W V + JV,W JU JU,W JV + JV, U JW} Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

22 Biharmonic submanifolds of CP n Biharmonic submanifolds of CP n Examples of biharmonic submanifolds in CP n Let i : Σ m N n (ρ) be a submanifold of real dimension m. (Gauss) N X Y = XY + σ(x,y) (Weingarten) N X V = A VX + X V Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

23 Biharmonic submanifolds of CP n Biharmonic submanifolds of CP n Examples of biharmonic submanifolds in CP n Let i : Σ m N n (ρ) be a submanifold of real dimension m. (Gauss) N X Y = XY + σ(x,y) (Weingarten) N X V = A VX + X V N n (ρ) = CP n (4) The biharmonic equation is τ (i) = m{ H + mh 3J (JH) } = 0 Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

24 Biharmonic submanifolds of CP n Examples of biharmonic submanifolds in CP n Biharmonic submanifolds of CP n Proposition If JH is tangent to Σ m, then Σ m is biharmonic iff { H + traceσ(,a H ( )) (m + 3)H = 0 4traceA ( ) H ( ) + mgrad( H ) = 0. Theorem (F. - Loubeau - Montaldo - Oniciuc, 010) If JH is tangent to Σ m and H = constant 0, then 1 If Σ m is proper-biharmonic, then H (0, m+3 m ]. If H = m+3 m, then Σm is proper-biharmonic iff it is pseudo umbilical, i.e., A H = H Id, and H = 0. Remark The upper bound of H is reached for curves. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

25 Biharmonic submanifolds of CP n Examples of biharmonic submanifolds in CP n Biharmonic submanifolds of CP n Proposition If JH is normal to Σ m, then Σ m is biharmonic if and only if { H + traceσ(,a H ( )) mh = 0 4traceA ( ) H ( ) + mgrad( H ) = 0. Moreover, if Σ m has parallel mean curvature, i.e., H = 0, then it is biharmonic iff traceσ(,a H ( )) = mh. Theorem (F. - Loubeau - Montaldo - Oniciuc, 010) If JH is normal to Σ m and H = constant 0, then 1 If Σ m is proper-biharmonic, then H (0,1]. If H = 1, then Σ m is proper-biharmonic iff it is pseudo-umbilical and H = 0. Remark The upper bound is reached for curves. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

26 Biharmonic submanifolds of CP n The Hopf fibration and the biharmonic equation The Hopf fibration and the biharmonic equation let π : C n+1 \ {0} CP n be the natural projection. the restriction to the sphere S n+1 C n+1 is the Hopf fibration π : S n+1 CP n Σ = π 1 (Σ m ) is the Hopf tube over Σ m CP n (4) ī Σ S n+1 π Σ i CP n Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

27 Biharmonic submanifolds of CP n The Hopf fibration and the biharmonic equation The Hopf fibration and the biharmonic equation Theorem (F.- Loubeau - Montaldo - Oniciuc, 010) Let i : Σ m CP n be an m-dimensional submanifold and ī : Σ = π 1 (Σ) S n+1 the corresponding Hopf-tube. Then we have (τ (i)) H = τ (ī) 4Ĵ(Ĵτ(ī)) + div((ĵτ(ī)) )ξ where X = X H is the horizontal lift with respect to the Hopf map, ξ is the Hopf vector field on S n+1 tangent to the fibres of the Hopf fibration, i.e., ξ (p) = Ĵp at any p S n+1, and Ĵ is the complex structure of R n+. Remark If Ĵτ(ī) is normal to Σ, then τ (i) = 0 iff τ (ī) = 0. If Ĵτ(ī) is tangent to Σ, then τ (i) = 0 and div Σ ((Jτ(i)) ) = 0 iff τ (ī) + 4τ(ī) = 0. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

28 Biharmonic submanifolds of CP n The Hopf fibration and the biharmonic equation The Hopf fibration and the biharmonic equation Theorem (Reckziegel, 1985) A totally real isometric immersion i : Σ m CP n (ρ) can be lifted locally (or globally, if Σ m is simply connected) to a horizontal immersion ĩ : Σ m S n+1 (ρ/4). Conversely, if ĩ : Σ m S n+1 (ρ/4) is a horizontal isometric immersion, then π(ĩ) : Σ m CP n (ρ) is a totally real isometric immersion. Moreover, we have π σ = σ, where σ and σ are the second fundamental forms of the immersions ĩ and i, respectively. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

29 Biharmonic submanifolds of CP n The Hopf fibration and the biharmonic equation The Hopf fibration and the biharmonic equation Theorem (Reckziegel, 1985) A totally real isometric immersion i : Σ m CP n (ρ) can be lifted locally (or globally, if Σ m is simply connected) to a horizontal immersion ĩ : Σ m S n+1 (ρ/4). Conversely, if ĩ : Σ m S n+1 (ρ/4) is a horizontal isometric immersion, then π(ĩ) : Σ m CP n (ρ) is a totally real isometric immersion. Moreover, we have π σ = σ, where σ and σ are the second fundamental forms of the immersions ĩ and i, respectively. Proposition (F.-Loubeau-Montaldo-Oniciuc, 010) Let ĩ : Σ m S n+1 (ρ/4) be a horizontal isometric immersion and consider the totally real isometric immersion i = π(ĩ) : Σ m CP n (ρ). Then (τ (i)) H = τ (ĩ) 4Ĵ(Ĵτ(ĩ)) + div Σ m ((Ĵτ(ĩ)) )ξ. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

30 Curves in CP n Biharmonic submanifolds of CP n Biharmonic pmc surfaces Definition A curve γ : I R CP n (ρ) parametrized by arc-length is called a Frenet curve of osculating order r, 1 r n, if there exist r orthonormal vector fields {E 1 = γ,...,e r } along γ such that CPn E 1 E 1 = κ 1 E CPn E 1 E i = κ i 1 E i 1 + κ i E i+1 CPn E 1... E r = κ r 1 E r 1 for all i {,...,r 1}, where {κ 1,κ,...,κ r 1 } are positive functions on I called the curvatures of γ. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

31 Curves in CP n Biharmonic submanifolds of CP n Biharmonic pmc surfaces a Frenet curve of osculating order r is called a helix of order r if κ i = constant > 0 for 1 i r 1. A helix of order is called a circle, and a helix of order 3 is called a helix the complex torsions τ ij of the curve γ are given by where 1 i < j r τ ij = E i,je j a helix of order r is called a holomorphic helix of order r if all its complex torsions are constant Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

32 Biharmonic submanifolds of CP n Biharmonic pmc surfaces The existence of holomorphic helices Theorem (Maeda-Adachi, 1997) For given positive constants κ 1, κ, and κ 3, there exist four equivalence classes of holomorphic helices of order 4 in CP (ρ) with curvatures κ 1, κ, and κ 3 with respect to holomorphic isometries of CP (ρ). Theorem (Maeda-Adachi, 1997) For any positive number κ and for any number τ, such that τ < 1, there exits a holomorphic circle with curvature κ and complex torsion τ in any complex space form. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

33 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Definition If the mean curvature vector field H of a surface Σ immersed in a complex space form is parallel in the normal bundle, i.e., H = 0, then Σ is called a pmc surface. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

34 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Definition If the mean curvature vector field H of a surface Σ immersed in a complex space form is parallel in the normal bundle, i.e., H = 0, then Σ is called a pmc surface. Theorem (F., 01) The (,0)-part Q (,0) of the quadratic form Q defined on a pmc surface Σ N n (ρ) by Q(X,Y) = 8 H A H X,Y + 3ρ X,T Y,T, where T = (JH) is the tangent part of JH, is holomorphic. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

35 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Theorem (F. - Pinheiro, 015) Let Σ be a complete non-minimal pmc surface with non-negative Gaussian curvature K isometrically immersed in a complex space form N n (ρ), ρ 0. Then one of the following holds: 1 the surface is flat; there exists a point p Σ such that K(p) > 0 and Q (,0) vanishes on Σ. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

36 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Sketch of the proof Consider a symmetric traceless operator S on Σ, given by S = 8 H A H + 3ρ T, T ( 3ρ T + 8 H 4) Id SX,Y = Q(X,Y) traceq X, Y Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

37 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Sketch of the proof Consider a symmetric traceless operator S on Σ, given by S = 8 H A H + 3ρ T, T ( 3ρ T + 8 H 4) Id SX,Y = Q(X,Y) traceq X, Y [Cheng-Yau, 1977] 1 S = K S + S 0 where K is the Gaussian curvature of the surface Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

38 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Sketch of the proof Consider a symmetric traceless operator S on Σ, given by S = 8 H A H + 3ρ T, T ( 3ρ T + 8 H 4) Id SX,Y = Q(X,Y) traceq X, Y [Cheng-Yau, 1977] 1 S = K S + S 0 where K is the Gaussian curvature of the surface K 0 ( Σ = parabolic) S = bounded Q.E.D. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

39 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Proposition Let Σ be a pmc surface in a complex space form (N(ρ),J,, ). Then Σ is biharmonic iff traceσ(,a H ) = ρ 4 {H 3(JT) } and (JT) = 0 where T is the tangent part of JH. Remark Proper-biharmonic pmc surfaces exist only in CP n (ρ), since 0 < A H = ρ 4 { H + 3 T } Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June 016 / 30

40 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Proposition (F. - Pinheiro, 015) If Σ is a proper-biharmonic pmc surface in CP n (ρ) then T = (JH) has constant length. Moreover, if T = constant 0, then T = 0. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

41 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Proposition (F. - Pinheiro, 015) If Σ is a proper-biharmonic pmc surface in CP n (ρ) then T = (JH) has constant length. Moreover, if T = constant 0, then T = 0. Proposition (F. - Pinheiro, 015) If Σ is a complete proper-biharmonic pmc surface in CP n (ρ) with K 0 and T = 0, then n 3 and Σ is pseudo-umbilical and totally real. Moreover, the mean curvature of Σ is H = ρ/. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

42 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Proposition (F. - Pinheiro, 015) If Σ is a proper-biharmonic pmc surface in CP n (ρ) then T = (JH) has constant length. Moreover, if T = constant 0, then T = 0. Proposition (F. - Pinheiro, 015) If Σ is a complete proper-biharmonic pmc surface in CP n (ρ) with K 0 and T = 0, then n 3 and Σ is pseudo-umbilical and totally real. Moreover, the mean curvature of Σ is H = ρ/. Proposition (F. - Pinheiro, 015) If Σ is a complete proper-biharmonic pmc surface in CP n (ρ) with K 0 and T 0, then the surface is flat and A H = 0. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

43 Biharmonic submanifolds of CP n Biharmonic pmc surfaces Biharmonic pmc surfaces in CP n (ρ) Theorem (Balmuş - Montaldo - Oniciuc, 008) Let Σ m be a proper-biharmonic cmc submanifold in S n (ρ/4) with mean curvature vector field H. Then H (0, ρ/] and, moreover, H = ρ/ if and only if Σ m is minimal in a small hypersphere S n 1 (ρ/) S n (ρ/4). Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

44 Biharmonic submanifolds of CP n The classification theorem Biharmonic pmc surfaces Theorem (F. - Pinheiro, 015) Let Σ be a complete proper-biharmonic pmc surface with non-negative Gaussian curvature in CP n (ρ). Then Σ is totally real and either 1 Σ is pseudo-umbilical and its mean curvature is equal to ρ/. Moreover, Σ = π( Σ ) CP n (ρ), n 3, where π : S n+1 (ρ/4) CP n (ρ) is the Hopf fibration and the horizontal lift Σ of Σ is a complete minimal surface in a small hypersphere S n (ρ/) S n+1 (ρ/4); or Σ lies in CP (ρ) as a complete Lagrangian proper-biharmonic pmc surface. Moreover, if ρ = 4, then Σ = π (S 1( 9± ) 41 0 S 1( 11 ) S 1( 11 )) CP (4); or 3 Σ lies in CP 3 (ρ) and Σ = γ 1 γ CP 3 (ρ), where γ 1 : R CP (ρ) CP 3 (ρ) is a holomorphic helix of order 4 with curvatures κ 1 = 7ρ 6, κ = 1 5ρ 4, κ 3 = 3 ρ 4 and complex torsions τ 1 = τ 34 = , τ 3 = τ 14 = 70 4, τ 13 = τ = 0, and γ : R CP 3 (ρ) is a circle with curvature κ = ρ/ and complex torsion τ 1 = 0. Moreover, γ 1 and γ always exist and are unique up to holomorphic isometries. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

45 Biharmonic submanifolds of CP n Biharmonic pmc surfaces The classification theorem (the proof) Case I: T = (JH) = 0 let π : S n+1 (ρ/4) CP n (ρ) be the Hopf fibration and Σ the horizontal lift of Σ Σ = proper-biharmonic Σ = pseudo-umbilical and totally real with H = ρ/ [Reckziegel, 1985] Σ S n+1 (ρ/4) is pseudo-umbilical and pmc [F. - L. - M. - O., 010] and T = 0 Σ = proper-biharmonic iff Σ = proper-biharmonic [Balmuş - Montaldo - Oniciuc, 008] Σ = minimal in a small hypersphere S n (ρ/) S n+1 (ρ/4) Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

46 Biharmonic submanifolds of CP n Biharmonic pmc surfaces The classification theorem (the proof) Case II: T = (JH) 0 Σ = proper-biharmonic Σ = totally real and flat with A H = 0 U = normal, U H, U J(TΣ ), Ricci eq. [A H,A U ] = 0 Σ pseudo-umbilical ( T = constant 0) A U = 0 consider the global orthonormal frame field {E 1 = T/ T,E } on Σ [F. - P., 015] E 1 = 0 and E = 0 if JH = tangent ( T = H ), then L = span{je 1,JE } NΣ is parallel ( L L), J(TΣ L) = TΣ L [Eschenburg - Tribuzy, 1993] Σ is a complete Lagrangian proper-biharmonic pmc surface in CP (ρ) ρ = 4: [Sasahara, 007] () Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

47 Biharmonic submanifolds of CP n Biharmonic pmc surfaces The classification theorem (the proof) if JH tangent ( T < H ), then L = span{e 3 = JE 1,E 4 = JE,E 5 = 1 N JN,E 6 = 1 N N } NΣ is parallel, J(TΣ L) = TΣ L (where N = (JH) ) [Eschenburg - Tribuzy, 1993] Σ lies in CP 3 (ρ) Σ = totally real, Ricci eq., trace(a H A U ) = (ρ/4){ H,U 3 JT,U }, K = 0 H = ρ 3 and A 3 = 1 ( ρ ), A 4 = 1 ( ρ ), A 5 = 1 ( 5ρ ), A 6 = 0 (1) E 1 = E = 0, de Rham Decomposition Theorem Σ = γ 1 γ (1) and [Maeda - Adachi, 1997] (3) Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

48 References References D. Fetcu, E. Loubeau, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of CP n, Math. Z. 66(010), D. Fetcu and A. L. Pinheiro, Biharmonic surfaces with parallel mean curvature in complex space forms, Kyoto J. Math. 55 (015), The Bibliography of Biharmonic Maps. Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30

49 Thank you Dorel Fetcu (TUIASI) Biharmonic pmc surfaces Varna, June / 30