Biconservative surfaces in Riemannian manifolds

Biconservative surfaces in Riemannian manifolds Simona Nistor Alexandru Ioan Cuza University of Iaşi Harmonic Maps Workshop Brest, May 15-18, 2017 1 / 55

Content 1 The motivation of the research topic 2 / 55

Content 1 The motivation of the research topic 2 Introducing the biconservative immersions 2 / 55

Content 1 The motivation of the research topic 2 Introducing the biconservative immersions 3 Biharmonic and biconservative submanifolds 2 / 55

Content 1 The motivation of the research topic 2 Introducing the biconservative immersions 3 Biharmonic and biconservative submanifolds 4 Biconservative surfaces in 3-dimensional space forms Local intrinsic characterization of biconservative surfaces in N 3 (c) Complete biconservative surfaces in R 3 Complete biconservative surfaces in S 3 2 / 55

General context The study of submanifolds with constant mean curvature, i.e., CMC submanifolds, and of minimal submanifolds, represents a very active research topic in Differential Geometry for more than 50 years. There are several ways to generalize these submanifolds: the study of CMC submanifolds which satisfy some additional geometric hypotheses (CMC + biharmonicity); the study of hypersurfaces in space forms, i.e., with constant sectional curvature, which are highly non-cmc. 4 / 55

General context Biconservative submanifolds in arbitrary manifolds (and in particular, biconservative surfaces) which are also CMC have some remarkable properties. 5 / 55

General context Biconservative submanifolds in arbitrary manifolds (and in particular, biconservative surfaces) which are also CMC have some remarkable properties. The CMC hypersurfaces in space forms are trivially biconservative, so more interesting is the study of biconservative hypersurfaces which are non-cmc. 5 / 55

Biharmonic maps Let (M m,g) and (N n,h) be two Riemannian manifolds. Assume that M is compact and consider Bienergy functional Euler-Lagrange equation E 2 : C (M,N) R, E 2 (φ) = 1 2 M τ(φ) 2 v g τ 2 (φ) = φ τ(φ) trace g R N (dφ,τ(φ))dφ = 0. Critical points of E 2 are called biharmonic maps. 7 / 55

The biharmonic equation (G.Y. Jiang, 1986) 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; 8 / 55

The stress-bienergy tensor G.Y. Jiang, 1987 defined the stress-energy tensor S 2 for the bienergy functional, and called it the stress-bienergy tensor: S 2 (X,Y) = 1 2 τ(φ) 2 X,Y + dφ, τ(φ) X,Y dφ(x), Y τ(φ) dφ(y), X τ(φ). It satisfies divs 2 = τ 2 (φ),dφ. φ = biharmonic divs 2 = 0. If φ is a submersion, divs 2 = 0 if and only if φ is biharmonic. 9 / 55

Biharmonic submanifolds; Biconservative submanifolds Definition A submanifold φ : M m N n is called biharmonic if φ is a biharmonic map, i.e., τ 2 (φ) = 0. 11 / 55

Biharmonic submanifolds; Biconservative submanifolds Definition A submanifold φ : M m N n is called biharmonic if φ is a biharmonic map, i.e., τ 2 (φ) = 0. Definition A submanifold φ : M m N n is called biconservative if divs 2 = 0, i.e., τ 2 (φ) = 0. 11 / 55

M m submanifold of N n

M m submanifold of N n M m biconservative

M m submanifold of N n M m biconservative M m biharmonic

M m submanifold of N n M m biconservative M m biharmonic M m minimal 12 / 55

Characterization results Theorem ([Loubeau, Montaldo, Oniciuc 2008]) A submanifold φ : M m N n is biharmonic if and only if tracea H ( ) + trace A H + trace ( R N (,H) )T = 0 and H + traceb(,a H ( )) + trace ( R N (,H) ) = 0, where H = traceb/m C(NM) is the mean curvature vector field. Proposition Let φ : M m N n be a submanifold. The following conditions are equivalent: 1 M is biconservative; 2 tracea H ( ) + trace A H + trace ( R N (,H) )T = 0; 3 m 2 grad ( H 2) + 2traceA H ( ) + 2trace( R N (,H) )T = 0; 4 2trace A H m 2 grad( H 2) = 0. 13 / 55

Examples of biconservative submanifolds Proposition Let φ : M m N n be a submanifold. If A H = 0, then M is biconservative. 14 / 55

Examples of biconservative submanifolds Proposition Let φ : M m N n be a submanifold. If A H = 0, then M is biconservative. Proposition Let φ : M m N n be a submanifold. If N is a space form, i.e., has constant sectional curvature, and M is PMC, i.e., has H parallel in NM, then M is biconservative. 14 / 55

Properties of biconservative submanifolds Proposition ([Balmuş, Montaldo, Oniciuc 2013]) Let φ : M m N n be a biconservative submanifold. Assume that M is pseudoumbilical, i.e., A H = H 2 I, and m 4. Then M is CMC. Proposition([N. 2017]) Let φ : M 2 N n be a CMC biconservative surface and assume that M is compact. If K 0, then A H = 0 and M is flat or pseudoumbilical. 15 / 55

Characterization theorems Theorem([Ou 2010]) If φ : M m N m+1 is a hypersurface, then M is biharmonic if and only if and 2A(gradf ) + f gradf 2f ( Ricci N (η) ) T = 0, f + f A 2 f Ricci N (η,η) = 0, where η is the unit normal vector field along M in N and f = tracea is the mean curvature function. A hypersurface φ : M m N m+1 (c) is biconservative if and only if A(gradf ) = f 2 gradf. 16 / 55

Biconservative surfaces in N 3 (c) Let φ : M 2 N 3 (c) be a non-cmc biconservative surface. Local results Global results

Biconservative surfaces in N 3 (c) Let φ : M 2 N 3 (c) be a non-cmc biconservative surface. extrinsic Local results Global results

Biconservative surfaces in N 3 (c) Let φ : M 2 N 3 (c) be a non-cmc biconservative surface. extrinsic extrinsic Local results Global results

Biconservative surfaces in N 3 (c) Let φ : M 2 N 3 (c) be a non-cmc biconservative surface. extrinsic extrinsic Local results Global results intrinsic

Biconservative surfaces in N 3 (c) Let φ : M 2 N 3 (c) be a non-cmc biconservative surface. extrinsic extrinsic Local results Global results intrinsic intrinsic 18 / 55

Biconservative surfaces in N 3 (c) extrinsic gradf 0 on M Local conditions intrinsic c K > 0 on M, gradk 0 on M, and the level curves of K are certain circles

Biconservative surfaces in N 3 (c) extrinsic gradf 0 on M Local conditions intrinsic c K > 0 on M, gradk 0 on M, and the level curves of K are certain circles Global conditions (M, g) complete and the above properties hold on an open and dense subset of M 19 / 55

Local results Theorem ([Caddeo, Montaldo, Oniciuc, Piu 2014]) Let φ : M 2 N 3 (c) be a biconservative surface with gradf 0 at any point of M. Then the Gaussian curvature K satisfies (i) the extrinsic condition K = deta + c = 3f 2 4 + c; (ii) the intrinsic conditions c K > 0, gradk 0 on M, and its level curves are circles in M with constant curvature κ = 3 gradk 8(c K) ; (iii) (c K) K gradk 2 8 3 K(c K)2 = 0, where is the Laplace-Beltrami operator on M. 20 / 55

Local intrinsic characterization Theorem ([Fetcu, N., Oniciuc 2016]) Let ( M 2,g ) be an abstract surface and c R a constant. Then, M can be locally isometrically embedded in N 3 (c) as a biconservative surface with gradf 0 at any point if and only if c K > 0, gradk 0, at any point, and its level curves are circles in M with constant curvature κ = 3 gradk 8(c K). If the surface M is simply connected, then the theorem holds globally, but, in this case, instead of a local isometric embedding we have a global isometric immersion. 22 / 55

Local intrinsic characterization Theorem ([N., Oniciuc 2017]) Let ( M 2,g ) be an abstract surface with Gaussian curvature K satisfying c K(p) > 0 and (gradk)(p) 0 at any point p M, where c R is a constant. Then, the level curves of K are circles in M with constant curvature κ = 3 gradk /(8(c K)) if and only if one of the following equivalent conditions holds (i) locally, g = e 2ρ ( du 2 + dv 2), ρ = ρ(u) satisfies and ρ > 0; ρ = e 2ρ/3 ce 2ρ ρ dτ u(ρ) = ρ 0 3e 2τ/3 ce 2τ + a + u 0, where ρ is in some open interval I, ρ 0 I and a,u 0 R; 23 / 55

Local intrinsic characterization Theorem ([N., Oniciuc 2017]) Let ( M 2,g ) be an abstract surface with Gaussian curvature K satisfying c K(p) > 0 and (gradk)(p) 0 at any point p M, where c R is a constant. Then, the level curves of K are circles in M with constant curvature κ = 3 gradk /(8(c K)) if and only if one of the following equivalent conditions holds (i) locally, g = e 2ρ ( du 2 + dv 2), ρ = ρ(u) satisfies (ii) locally, g = e 2ρ ( du 2 + dv 2), ρ = ρ(u) satisfies and ρ > 0; ρ = e 2ρ/3 ce 2ρ ρ dτ u(ρ) = ρ 0 3e 2τ/3 ce 2τ + a + u 0, where ρ is in some open interval I, ρ 0 I and a,u 0 R; 3ρ + 2ρ ρ + 8ce 2ρ ρ = 0, ρ > 0 and c + e 2ρ ρ > 0; ρ dτ u(ρ) = ρ 0 3be 2τ/3 ce 2τ + a +u 0, where ρ is in some open interval I, ρ 0 I and a,b,u 0 R, b > 0. 23 / 55

Local extrinsic results in R 3 25 / 55

Local extrinsic result in R 3 Theorem ([Hasanis, Vlachos 1995]) Let M 2 be a surface in R 3 with (gradf )(p) 0 at any p M. Then, M 2 is biconservative if and only if, locally, it is a surface of revolution, and the curvature κ = κ(u) of the profile curve σ = σ(u), σ (u) = 1, is positive solution of the following ODE κ κ = 7 4 ( κ ) 2 4κ 4. 26 / 55

Local extrinsic result in R 3 Theorem ([Caddeo, Montaldo, Oniciuc, Piu 2014]) Let M 2 be a biconservative surface in R 3 with (gradf )(p) 0 at any p M. Then, locally, the surface can be parametrized by ( ) X C 0 (θ,v) = θ cosv,θ sinv,u C 0 (θ), where u C 0 (θ) = 3 2 C 0 ( θ 1/3 C 0 θ 2/3 1 + 1 ( ) ) log C 0 θ 1/3 + C 0 θ 2/3 1 C 0 ( ) with C 0 a positive constant and θ C 3/2 0,. (( ) ) We denote X C 0 C 3/2 0, R = S C 0. 27 / 55

Global extrinsic results in R 3 28 / 55

Global extrinsic result in R 3 Proposition ([Montaldo, Oniciuc, Ratto 2016, N. 2016]) If we consider the symmetry of Grafu C 0, with respect to the Oθ(= Ox) axis, we get a smooth, complete, biconservative surface S C 0 in R 3. Moreover, its mean curvature function has its gradient grad f C 0 is different from zero at any point of an open dense subset of S C 0. 29 / 55

S C 0 30 / 55

S C 0 S C 0 30 / 55

Local intrinsic results corresponding to c = 0 31 / 55

Local intrinsic result; c = 0 Proposition ([N. 2016]) Let ( M 2,g = e 2ρ ( du 2 + dv 2)) an abstract surface. Then g satisfies the local intrinisic conditions with c = 0 if and only if where C 0 > 0 is a constant. g C0 = C 0 (coshu) 6 ( du 2 + dv 2), 32 / 55

Global intrinsic results corresponding to c = 0 33 / 55

Global intrinsic result; c = 0 Theorem ([N. 2016]) ( Let R 2,g C0 = C 0 (coshu) 6 ( du 2 + dv 2)). Then, we have: 34 / 55

Global intrinsic result; c = 0 Theorem ([N. 2016]) ( Let R 2,g C0 = C 0 (coshu) 6 ( du 2 + dv 2)). Then, we have: (i) ( R 2,g C0 ) is complete; (ii) the immersion φ C0 : ( R 2,g C0 ) R 3 given by φ C0 (u,v) = ( σ 1 C 0 (u)cos(3v),σ 1 C 0 (u)sin(3v),σ 2 C 0 (u) ) is biconservative in R 3, where σc 1 C0 0 (u) = 3 (coshu)3, σc 2 0 (u) = C0 2 ( ) 1 2 sinh(2u) + u, u R. 34 / 55

Uniqueness Theorem ([N., Oniciuc 2017]) Let M 2 be a biconservative regular surface in R 3. If M is compact, then M is CMC. 35 / 55

Uniqueness Theorem ([N., Oniciuc 2017]) Let M 2 be a biconservative regular surface in R 3. If M is compact, then M is CMC. Theorem Let M 2 be a biconservative, complete and non-compact regular surface in R 3. Then M = S C 0. 35 / 55

Local extrinsic results in S 3 37 / 55

Local extrinsic result in S 3 Theorem ([Caddeo, Montaldo, Oniciuc, Piu 2014]) Let M 2 be a biconservative surface in S 3 with (gradf )(p) 0 for any p M. Then, locally, the surface viewed in R 4, can be parametrized by ( ( ) 2 ( ) 2 4 Y C 1 (θ,v) = 1 θ 3/4 cos µ(θ), 1 4 θ 3/4 sin µ(θ), 3 C 1 3 C 1 ) 4 θ 3/4 cosv, 4 θ 3/4 sinv 3 C 1 3 C 1, (1) where (θ,v) (θ 01,θ 02 ) R, θ 01 and θ 02 are positive solutions of the equation 16 9 θ 2 16θ 4 + C 1 θ 7/2 = 0 and µ(θ) = ± θ θ 0 E(τ) dτ + c k, with c k R, k Z, and θ 0 (θ 01,θ 02 ). If k = 0, we denote by S C 1 = Y C 1 ((θ 01,θ 02 ) R). 38 / 55

Global extrinsic results in S 3 39 / 55

Global extrinsic result in S 3 The idea of the construction is to start with a surface S C 1 and then to consider ( T k S C 1 ), where T k is a linear orthogonal transformation of R 4 that acts on span{e 1,e 2 } as an axial orthogonal symmetry and leaves invariant span{e 3,e 4 }, for k Z. We perform it infinitely many times. 40 / 55

Using the stereographic projection, this construction can be illustrated in R3. 41 / 55

Using the stereographic projection, this construction can be illustrated in R3. N(0, 0, 0, 1) k { 2, 1, 0, 1, 2} 41 / 55

Using the stereographic projection, this construction can be illustrated in R3. N(0, 0, 0, 1) k { 2, 1, 0, 1, 2} N 0 (1, 0, 0, 0) k { 2, 1, 0, 1, 2} 41 / 55

Local intrinsic results corresponding to c = 1 42 / 55

Local intrinsic result; c = 1 Proposition ([N. 2016]) Let ( M 2,g ) be an abstract surface with g = e 2ρ(u) (du 2 + dv 2 ), where u = u(ρ) satisfies ρ dτ u = ρ 0 3be 2τ/3 e 2τ + a + u 0, where ρ is in some open interval I, a,b R are positive constants, and u 0 R is a constant. Then ( M 2,g ) is isometric to ( D C1,g C1 = ) 3 ξ ( 2 ξ 8/3 + 3C 1 ξ 2 3 )dξ 2 + 1 ξ 2 dθ 2, where D C1 = (ξ 01,ξ 02 ) R, C 1 ( 4/ ( 3 3/2), ) is a positive constant, and ξ 01 and ξ 02 are the positive vanishing points of ξ 8/3 + 3C 1 ξ 2 3, with 0 < ξ 01 < ξ 02. 43 / 55

Theorem ([N. 2016]) Let ( D C1,g C1 ). Then 44 / 55

Theorem ([N. 2016]) Let ( D C1,g C1 ). Then (i) ( D C1,g C1 ) is not complete; 44 / 55

Theorem ([N. 2016]) Let ( D C1,g C1 ). Then (i) ( D C1,g C1 ) is not complete; (ii) the immersion φ C1 : ( D C1,g C1 ) S 3 given by φ C1 (ξ,θ) = ( 1 1 cosζ (ξ ), C 1 ξ 2 1 1 ) cos( C1 θ) sinζ (ξ ),, sin( C 1 θ), C 1 ξ 2 C1 ξ C1 ξ is biconservative in S 3, where ζ (ξ ) = ± ξ ξ 00 E(τ) dτ + c k, with c k R, k Z, and ξ 00 (ξ 01,ξ 02 ). 44 / 55

Global intrinsic results corresponding to c = 1 45 / 55

The key ingredient Theorem Let ( ) ( ) D C1,g C1. Then DC1,g C1 is the universal cover of the surface of revolution in R 3 given by ( ψ C1,C1 (ξ,θ) = χ(ξ )cos θ C1, χ(ξ )sin θ ) C1,ν(ξ ), (2) where χ(ξ ) = C1 /ξ,ν(ξ ) = ± ξ E(τ) dτ + c ξ 00 k, C 1 (0, ( C 1 4/3 3/2) ) 1/2 is a positive constant and c k R, k Z. 46 / 55

θ ISOMETRY ξ 01 ξ 02 ξ ( M 2,g ) (D C1,g C1 )

θ ISOMETRY ξ 01 ξ 02 ξ ( M 2,g ) (D C1,g C1 ) BICONSERVATIVE φ C1 = φ ± C 1,c k S 3

θ ISOMETRY ξ 01 ξ 02 ξ ( M 2,g ) (D C1,g C1 ) ISOMETRY ψ C1,C 1 = ψ± C 1,C 1,c k S ± C 1,C 1,c 1 R 3 φ C1 = φ ± C 1,c k BICONSERVATIVE S 3

θ ISOMETRY ξ 01 ξ 02 ξ ( M 2,g ) (D C1,g C1 ) ISOMETRY ψ C1,C 1 = ψ± C 1,C 1,c k S ± C 1,C 1,c 1 R 3 φ C1 = φ ± C 1,c k BICONSERVATIVE playing with the const. c k and ± S 3 S C1,C 1 R3 complete

θ ISOMETRY ξ 01 ξ 02 ξ ( M 2,g ) (D C1,g C1 ) ISOMETRY ψ C1,C 1 = ψ± C 1,C 1,c k S ± C 1,C 1,c 1 R 3 playing with the const. c k and ± playing with the const. c k and ±, Φ C1,C 1 φ C1 = φ ± C 1,c k BICONSERVATIVE S 3 S C1,C 1 R3 complete 47 / 55

The projection of Φ C1,C1 on the Ox1 x 2 plane is a curve which lies in the annulus of radii 1 1/ ( C 1 ξ01 2 ) and 1 1/ ( C 1 ξ02) 2. It has self-intersections and is dense in the annulus. Choosing C 1 = C1 = 1, we obtain x 2 x 1 48 / 55

The signed curvature of the profile curve of S C1,C 1. κ ν 49 / 55

The signed curvature of the curve obtained projecting Φ 1,1 on the Ox 1 x 2 plane. κ ν 50 / 55

Open problem Let φ : M 2 N 3 (c) be a biconservative surface. If φ is CMC on an open subset of M, then φ is CMC on M. 51 / 55

Open problem Let φ : M 2 N 3 (c) be a biconservative surface. If φ is CMC on an open subset of M, then φ is CMC on M. φ : M 2 N 3 (c) is a biconservative surface if and only if A(gradf ) = f 2 gradf ; 51 / 55

Open problem Let φ : M 2 N 3 (c) be a biconservative surface. If φ is CMC on an open subset of M, then φ is CMC on M. φ : M 2 N 3 (c) is a biconservative surface if and only if A(gradf ) = f 2 gradf ; if φ : M 2 N 3 (c) is a biconservative surface, then f f 3 gradf 2 2 A,Hessf = 0. 51 / 55

References I [Balmuş, Montaldo, Oniciuc 2013] A. Balmuş, S. Montaldo, C. Oniciuc, Biharmonic PNMC submanifolds in spheres, Ark. Mat. 51 (2013), 197 221. [Caddeo, Montaldo, Oniciuc, Piu 2014] R. Caddeo, S. Montaldo, C. Oniciuc, P. Piu, Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor, Ann. Mat. Pura Appl. (4) 193 (2014), 529 550. [Fetcu, N., Oniciuc 2016] D. Fetcu, S. Nistor, C. Oniciuc, On biconservative surfaces in 3-dimensional space forms, Comm. Anal. Geom. (5) 24 (2016), 1027 1045. [Hasanis, Vlachos 1995] Th. Hasanis, Th. Vlachos, Hypersurfaces in E 4 with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145 169. [Loubeau, Montaldo, Oniciuc 2008] E. Loubeau, S. Montaldo, C. Oniciuc, The stress-energy tensor for biharmonic maps, Math. Z. 259 (2008), 503 524. 52 / 55

References II [Montaldo, Oniciuc, Ratto 2016] S. Montaldo, C. Oniciuc, A. Ratto, Biconservative Surfaces, J. Geom. Anal. 26 (2016), 313 329. [Montaldo, Oniciuc, Ratto 2016] S. Montaldo, C. Oniciuc, A. Ratto, Proper biconservative immersions into the Euclidean space, Ann. Mat. Pura Appl. (4) 195 (2016), 403 422. [N. 2016] S. Nistor, Complete biconservative surfaces in R 3 and S 3, J. Geom. Phys. 110 (2016) 130 153. [N. 2017] S. Nistor, On biconservative surfaces, preprint, arxiv: 1704.04598. [N., Oniciuc 2017] S. Nistor, C. Oniciuc Global properties of biconservative surfaces in R 3 and S 3, preprint, arxiv: 1701.07706. 53 / 55

References III [N., Oniciuc 2017] S. Nistor, C. Oniciuc On the uniqueness of complete biconservative surfaces in R 3, work in progress. [Ou 2010] Y.-L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math. 248 (2010), 217 232. 54 / 55

Thank you! 55 / 55