Characterization results Theorem ([Loubeau, Montaldo, Oniciuc – 2008]) A submanifold φ : M m → N n is biharmonic if and only if � T = 0 R N ( · , H ) · � trace A ∇ ⊥ · H ( · )+ trace ∇ A H + trace and � ⊥ = 0 , ∆ ⊥ H + trace B ( · , A H ( · ))+ trace R N ( · , H ) · � where H = trace B / m ∈ C ( NM ) is the mean curvature vector field. Proposition Let φ : M m → N n be a submanifold. The following conditions are equivalent: M is biconservative; 1 � T = 0 ; � R N ( · , H ) · · H ( · )+ trace ∇ A H + trace trace A ∇ ⊥ 2 � T = 0 ; m � | H | 2 � � R N ( · , H ) · + 2trace A ∇ ⊥ · H ( · )+ 2trace 2 grad 3 2trace ∇ A H − m | H | 2 � � 2 grad = 0 . 4 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 ([Balmu¸ s, 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 . 15 / 55
Properties of biconservative submanifolds Proposition ([Balmu¸ s, 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
Properties of biconservative submanifolds Proposition ([Balmu¸ s, 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. Proposition ([Montaldo, Oniciuc, Ratto – 2016]) Let φ : M 2 → N n be a biconservative surface. Then � A H ( ∂ z ) , ∂ z � is holomorphic if and only if M is CMC . 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 � T = 0 , Ricci N ( η ) � 2 A ( grad f )+ f grad f − 2 f and ∆ f + f | A | 2 − f Ricci N ( η , η ) = 0 , where η is the unit normal vector field along M in N and f = trace A is the mean curvature function. 16 / 55
Characterization theorems Theorem([Ou – 2010]) If φ : M m → N m + 1 is a hypersurface, then M is biharmonic if and only if � T = 0 , Ricci N ( η ) � 2 A ( grad f )+ f grad f − 2 f and ∆ f + f | A | 2 − f Ricci N ( η , η ) = 0 , where η is the unit normal vector field along M in N and f = trace A is the mean curvature function. A hypersurface φ : M m → N m + 1 ( c ) is biconservative if and only if A ( grad f ) = − f 2 grad f . 16 / 55
Characterization theorems Theorem([Ou – 2010]) If φ : M m → N m + 1 is a hypersurface, then M is biharmonic if and only if � T = 0 , Ricci N ( η ) � 2 A ( grad f )+ f grad f − 2 f and ∆ f + f | A | 2 − f Ricci N ( η , η ) = 0 , where η is the unit normal vector field along M in N and f = trace A is the mean curvature function. A hypersurface φ : M m → N m + 1 ( c ) is biconservative if and only if A ( grad f ) = − f 2 grad f . Every CMC hypersurface in N m + 1 ( c ) is biconservative. 16 / 55
Content The motivation of the research topic 1 Introducing the biconservative immersions 2 Biharmonic and biconservative submanifolds 3 Biconservative surfaces in 3 -dimensional space forms 4 Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces in R 3 Complete biconservative surfaces in S 3 17 / 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 ) grad f � = 0 on M extrinsic Local conditions c − K > 0 on M , grad K � = 0 on M , intrinsic and the level curves of K are certain circles
Biconservative surfaces in N 3 ( c ) grad f � = 0 on M extrinsic Local conditions c − K > 0 on M , grad K � = 0 on M , intrinsic and the level curves of K are certain circles ( M , g ) complete and the above properties Global conditions 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 grad f � = 0 at any point of M . Then the Gaussian curvature K satisfies (i) the extrinsic condition K = det A + c = − 3 f 2 4 + c ; (ii) the intrinsic conditions c − K > 0 , grad K � = 0 on M , and its level curves are circles in M with constant curvature κ = 3 | grad K | 8 ( c − K ) ; (iii) ( c − K ) ∆ K −| grad K | 2 − 8 3 K ( c − K ) 2 = 0 , where ∆ is the Laplace-Beltrami operator on M . 20 / 55
Content The motivation of the research topic 1 Introducing the biconservative immersions 2 Biharmonic and biconservative submanifolds 3 Biconservative surfaces in 3 -dimensional space forms 4 Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces in R 3 Complete biconservative surfaces in S 3 21 / 55
Local intrinsic characterization Theorem ([Fetcu, N., Oniciuc – 2016]) � M 2 , g � Let 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 grad f � = 0 at any point if and only if c − K > 0 , grad K � = 0 , at any point, and its level curves are circles in M with constant curvature κ = 3 | grad K | 8 ( c − K ) . 22 / 55
Local intrinsic characterization Theorem ([Fetcu, N., Oniciuc – 2016]) � M 2 , g � Let 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 grad f � = 0 at any point if and only if c − K > 0 , grad K � = 0 , at any point, and its level curves are circles in M with constant curvature κ = 3 | grad K | 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 ([Fetcu, N., Oniciuc – 2016]) � M 2 , g � Let 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 grad f � = 0 at any point if and only if c − K > 0 , grad K � = 0 , at any point, and its level curves are circles in M with constant curvature κ = 3 | grad K | 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. We remark that unlike in the minimal immersions case, if M satisfies the hypotheses from above theorem, then there exists a unique biconservative immersion in N 3 ( c ) (up to an isometry of N 3 ( c ) ), and not a one-parameter family. 22 / 55
Local intrinsic characterization Theorem ([N., Oniciuc – 2017]) � M 2 , g � Let be an abstract surface with Gaussian curvature K satisfying c − K ( p ) > 0 and ( grad K )( 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 | grad K | / ( 8 ( c − K )) if and only if one of the following equivalent conditions holds 23 / 55
Local intrinsic characterization Theorem ([N., Oniciuc – 2017]) � M 2 , g � Let be an abstract surface with Gaussian curvature K satisfying c − K ( p ) > 0 and ( grad K )( 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 | grad K | / ( 8 ( c − K )) if and only if one of the following equivalent conditions holds du 2 + dv 2 � (i) locally, g = e 2 ρ � , ρ = ρ ( u ) satisfies ρ ′′ = e − 2 ρ / 3 − ce 2 ρ and ρ ′ > 0 ; � ρ d τ u ( ρ ) = + u 0 , � − 3 e − 2 τ / 3 − ce 2 τ + a ρ 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]) � M 2 , g � Let be an abstract surface with Gaussian curvature K satisfying c − K ( p ) > 0 and ( grad K )( 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 | grad K | / ( 8 ( c − K )) if and only if one of the following equivalent conditions holds du 2 + dv 2 � du 2 + dv 2 � (i) locally, g = e 2 ρ � (ii) locally, g = e 2 ρ � , , ρ = ρ ( u ) satisfies ρ = ρ ( u ) satisfies 3 ρ ′′′ + 2 ρ ′ ρ ′′ + 8 ce 2 ρ ρ ′ = 0 , ρ ′′ = e − 2 ρ / 3 − ce 2 ρ and ρ ′ > 0 ; ρ ′ > 0 and c + e − 2 ρ ρ ′′ > 0 ; � ρ � ρ d τ d τ u ( ρ ) = + u 0 , u ( ρ ) = + u 0 , � − 3 be − 2 τ / 3 − ce 2 τ + a � − 3 e − 2 τ / 3 − ce 2 τ + a ρ 0 ρ 0 where ρ is in some open interval where ρ is in some open interval I , ρ 0 ∈ I and a , u 0 ∈ R ; I , ρ 0 ∈ I and a , b , u 0 ∈ R , b > 0 . 23 / 55
Content The motivation of the research topic 1 Introducing the biconservative immersions 2 Biharmonic and biconservative submanifolds 3 Biconservative surfaces in 3 -dimensional space forms 4 Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces in R 3 Complete biconservative surfaces in S 3 24 / 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 ( grad f )( 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 κ ′ � 2 − 4 κ 4 . � 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 ( grad f )( p ) � = 0 at any p ∈ M . Then, locally, the surface can be parametrized by � � C 0 ( θ , v ) = θ cos v , θ sin v , u ˜ C 0 ( θ ) , X ˜ where � �� �� 3 � 1 � C 0 θ 1 / 3 + C 0 θ 2 / 3 − 1 + C 0 θ 2 / 3 − 1 θ 1 / 3 ˜ ˜ ˜ C 0 ( θ ) = � ˜ u ˜ log 2 ˜ C 0 C 0 � � C − 3 / 2 with ˜ ˜ C 0 a positive constant and θ ∈ , ∞ . 0 �� � � C − 3 / 2 ˜ We denote X ˜ , ∞ × R = S ˜ C 0 . C 0 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 Graf u ˜ C 0 , with respect to the O θ (= Ox ) axis, we get a smooth, complete, biconservative surface ˜ C 0 in R 3 . Moreover, its mean S ˜ curvature function has its gradient grad ˜ C 0 is different from zero at any point of f ˜ an open dense subset of ˜ C 0 . S ˜ 29 / 55
S ˜ C 0 30 / 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]) du 2 + dv 2 �� M 2 , g = e 2 ρ � � Let an abstract surface. Then g satisfies the local intrinisic conditions with c = 0 if and only if du 2 + dv 2 � g C 0 = C 0 ( cosh u ) 6 � , where C 0 > 0 is a constant. 32 / 55
Global intrinsic results corresponding to c = 0 33 / 55
Global intrinsic result; c = 0 Theorem ([N. – 2016]) � du 2 + dv 2 �� R 2 , g C 0 = C 0 ( cosh u ) 6 � Let . Then, we have: 34 / 55
Global intrinsic result; c = 0 Theorem ([N. – 2016]) � du 2 + dv 2 �� R 2 , g C 0 = C 0 ( cosh u ) 6 � Let . Then, we have: � R 2 , g C 0 � (i) is complete; → R 3 given by � R 2 , g C 0 � (ii) the immersion φ C 0 : σ 1 C 0 ( u ) cos ( 3 v ) , σ 1 C 0 ( u ) sin ( 3 v ) , σ 2 � � φ C 0 ( u , v ) = C 0 ( u ) is biconservative in R 3 , where √ C 0 √ C 0 � 1 � ( cosh u ) 3 , σ 1 σ 2 C 0 ( u ) = C 0 ( u ) = 2 sinh ( 2 u )+ u , u ∈ R . 3 2 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 = ˜ C 0 . S ˜ 35 / 55
Content The motivation of the research topic 1 Introducing the biconservative immersions 2 Biharmonic and biconservative submanifolds 3 Biconservative surfaces in 3 -dimensional space forms 4 Local intrinsic characterization of biconservative surfaces in N 3 ( c ) Complete biconservative surfaces in R 3 Complete biconservative surfaces in S 3 36 / 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 ( grad f )( p ) � = 0 for any p ∈ M . Then, locally, the surface viewed in R 4 , can be parametrized by �� � � 2 � 2 � � 4 C 1 θ − 3 / 4 4 C 1 θ − 3 / 4 C 1 ( θ , v ) = 3 √ ˜ cos µ ( θ ) , 3 √ ˜ sin µ ( θ ) , Y ˜ 1 − 1 − (1) � C 1 θ − 3 / 4 cos v , C 1 θ − 3 / 4 sin v 3 √ ˜ 4 3 √ ˜ 4 , where ( θ , v ) ∈ ( θ 01 , θ 02 ) × R , θ 01 and θ 02 are positive solutions of the equation − 16 9 θ 2 − 16 θ 4 + ˜ C 1 θ 7 / 2 = 0 � θ and µ ( θ ) = ± θ 0 E ( τ ) d τ + ˜ c k ∈ R , k ∈ Z , and θ 0 ∈ ( θ 01 , θ 02 ) . c k , with ˜ 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 � � , where T k is a linear orthogonal transformation of R 4 that acts on T k S ˜ C 1 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 R 3 . 41 / 55
Using the stereographic projection, this construction can be illustrated in R 3 . N ( 0 , 0 , 0 , 1 ) k ∈ {− 2 , − 1 , 0 , 1 , 2 } 41 / 55
Using the stereographic projection, this construction can be illustrated in R 3 . N ′ ( 1 , 0 , 0 , 0 ) N ( 0 , 0 , 0 , 1 ) k ∈ {− 2 , − 1 , 0 , 1 , 2 } 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]) be an abstract surface with g = e 2 ρ ( u ) ( du 2 + dv 2 ) , where u = u ( ρ ) � M 2 , g � Let satisfies � ρ d τ u = + u 0 , � − 3 be − 2 τ / 3 − e 2 τ + a ρ 0 where ρ is in some open interval I , a , b ∈ R are positive constants, and u 0 ∈ R � M 2 , g � is a constant. Then is isometric to � � 3 � d ξ 2 + 1 ξ 2 d θ 2 D C 1 , g C 1 = , − ξ 8 / 3 + 3 C 1 ξ 2 − 3 ξ 2 � � � 3 3 / 2 � � where D C 1 = ( ξ 01 , ξ 02 ) × R , C 1 ∈ 4 / , ∞ is a positive constant, and ξ 01 and ξ 02 are the positive vanishing points of − ξ 8 / 3 + 3 C 1 ξ 2 − 3 , with 0 < ξ 01 < ξ 02 . 43 / 55
Theorem ([N. – 2016]) � � Let D C 1 , g C 1 . Then 44 / 55
Theorem ([N. – 2016]) � � Let D C 1 , g C 1 . Then � � (i) D C 1 , g C 1 is not complete; 44 / 55
Theorem ([N. – 2016]) � � Let D C 1 , g C 1 . Then � � (i) D C 1 , g C 1 is not complete; → S 3 given by � � (ii) the immersion φ C 1 : D C 1 , g C 1 C 1 ξ 2 sin ζ ( ξ ) , cos ( √ C 1 θ ) , sin ( √ C 1 θ ) �� � � 1 1 φ C 1 ( ξ , θ ) = 1 − C 1 ξ 2 cos ζ ( ξ ) , 1 − √ C 1 ξ √ C 1 ξ , � ξ is biconservative in S 3 , where ζ ( ξ ) = ± ξ 00 E ( τ ) d τ + 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 C 1 , g C 1 . Then D C 1 , g C 1 is the universal cover of the surface of revolution in R 3 given by � χ ( ξ ) cos θ , χ ( ξ ) sin θ � ψ C 1 , C ∗ 1 ( ξ , θ ) = , ν ( ξ ) , (2) C ∗ C ∗ 1 1 � ξ � C 1 − 4 / 3 3 / 2 � − 1 / 2 � where χ ( ξ ) = C ∗ ξ 00 E ( τ ) d τ + c ∗ k , C ∗ � 1 / ξ , ν ( ξ ) = ± 1 ∈ 0 , is a positive constant and c ∗ k ∈ R , k ∈ Z . 46 / 55
θ ISOMETRY ξ 01 ξ 02 ξ ( D C 1 , g C 1 ) M 2 , g � �
θ ISOMETRY ξ 01 ξ 02 ξ ( D C 1 , g C 1 ) M 2 , g � � BICONSERVATIVE φ C 1 = φ ± C 1 , ck S 3
θ ISOMETRY ξ 01 ξ 02 ξ ( D C 1 , g C 1 ) M 2 , g � � ISOMETRY = ψ ± ψ C 1 , C ∗ C 1 , C ∗ 1 , c ∗ 1 k BICONSERVATIVE φ C 1 = φ ± S ± 1 ⊂ R 3 C 1 , C ∗ 1 , c ∗ C 1 , ck S 3
θ ISOMETRY ξ 01 ξ 02 ξ ( D C 1 , g C 1 ) M 2 , g � � ISOMETRY = ψ ± ψ C 1 , C ∗ C 1 , C ∗ 1 , c ∗ 1 k BICONSERVATIVE φ C 1 = φ ± S ± 1 ⊂ R 3 C 1 , C ∗ 1 , c ∗ C 1 , ck p l a y i n g w i t h t h e ∗ c c o k a n n s d t . ± S 3 1 ⊂ R 3 complete ˜ S C 1 , C ∗
θ ISOMETRY ξ 01 ξ 02 ξ ( D C 1 , g C 1 ) M 2 , g � � ISOMETRY = ψ ± ψ C 1 , C ∗ C 1 , C ∗ 1 , c ∗ 1 k BICONSERVATIVE φ C 1 = φ ± S ± 1 ⊂ R 3 C 1 , C ∗ 1 , c ∗ C 1 , ck playing with p l a y i n the const. c k and ± , g w i t h t h e Φ C 1 , C ∗ ∗ c c o k a n n s 1 d t . ± S 3 1 ⊂ R 3 complete ˜ S C 1 , C ∗ 47 / 55
1 on the Ox 1 x 2 plane is a curve which lies in the The projection of Φ C 1 , C ∗ � � � C 1 ξ 2 � � C 1 ξ 2 � 1 − 1 / 1 − 1 / annulus of radii and . It has self-intersections 01 02 and is dense in the annulus. Choosing C 1 = C ∗ 1 = 1 , we obtain x 2 x 1 48 / 55
The signed curvature of the profile curve of ˜ 1 . S C 1 , C ∗ 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 ( grad f ) = − f 2 grad f ; 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 ( grad f ) = − f 2 grad f ; if φ : M 2 → N 3 ( c ) is a biconservative surface, then f ∆ f − 3 | grad f | 2 − 2 � A , Hess f � = 0 . 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 ( grad f ) = − f 2 grad f ; if φ : M 2 → N 3 ( c ) is a biconservative surface, then f ∆ f − 3 | grad f | 2 − 2 � A , Hess f � = 0 . Open problem Let M 2 be a biconservative regular surface in S 3 . If M is compact, then is M a CMC surface? 51 / 55
References I [Balmu¸ s, Montaldo, Oniciuc – 2013] A. Balmu¸ s, 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
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