On biconservative submanifolds Simona NistorBarna Alexandru Ioan - - PowerPoint PPT Presentation

On biconservative submanifolds

Simona Nistor–Barna

“Alexandru Ioan Cuza” University of Ia¸ si

Università degli Studi di Cagliari, April 6, 2017

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Content

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The motivation of the research topic General context Harmonic maps Biharmonic maps

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

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Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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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.

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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. Examples of minimal surfaces The plane The helicoid Enneper’s surface The catenoid

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General context

Examples of CMC surfaces The sphere The cylinder The nodoid The unduloid

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General context

There are several ways to generalize these submanifolds:

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General context

There are several ways to generalize these submanifolds: the study of CMC submanifolds which satisfy some additional geometric hypotheses (CMC + biharmonicity);

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General context

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”.

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General context

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”. The study of biconservative surfaces matches with both directions from above.

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General context

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

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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.

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General context

The biconservative surfaces are closely related to the biharmonic submanifolds.

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General context

The biconservative surfaces are closely related to the biharmonic submanifolds. The biharmonic submanifolds represent a particular case of biharmonic maps, they being defined by Riemannian immersions which are also biharmonic maps.

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General context

The biconservative surfaces are closely related to the biharmonic submanifolds. The biharmonic submanifolds represent a particular case of biharmonic maps, they being defined by Riemannian immersions which are also biharmonic maps. The biharmonic maps are critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 1 2

• M τ(φ)2 vg,

where τ(φ) = traceg ∇dφ is the tension field associated to φ, and its vanishing characterizes harmonic maps.

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Harmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume that M is compact and consider The energy functional E : C∞(M,N) → R, E(φ) = E1 (φ) = 1 2

• M dφ2 vg.

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Harmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume that M is compact and consider The energy functional E : C∞(M,N) → R, E(φ) = E1 (φ) = 1 2

• M dφ2 vg.

The harmonic maps are critical points of E, i.e., for any variation {φt}t∈R of φ we have d dt

• t=0

{E(φt)} = 0.

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The first variation of the energy functional

Theorem

A smooth map φ : (Mm,g) → (Nn,h) is harmonic if and only if the tension field associated to φ, τ(φ) = traceg∇dφ, vanishes.

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The first variation of the energy functional

Theorem

A smooth map φ : (Mm,g) → (Nn,h) is harmonic if and only if the tension field associated to φ, τ(φ) = traceg∇dφ, vanishes.

The expression of the tension field in local charts

τ(φ) = gij

• ∂ 2φ α

∂xi∂xj − MΓk

ij

∂φ α ∂xk + NΓα

βδ

∂φ β ∂xi ∂φ δ ∂xj ∂ ∂yα ◦φ

• ,

where Γ represent the Christoffel symbols.

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Harmonic maps

Example

Let Mm be a submanifold of a Riemannian manifold (Nn,h), i.e., φ : (Mm,g) → (Nn,h) is a Riemannian immersion. Then φ is a harmonic map if and only if M is a minimal submanifold.

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The stress-energy tensor

• D. Hilbert, 1924, ([10]), associated to a functional E a symmetric tensor

field S of type (1,1), or (0,2), which is conservative at the critical points of E, i.e., divS = 0 at these critical points, and called it the stress-energy tensor.

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The stress-energy tensor

• D. Hilbert, 1924, ([10]), associated to a functional E a symmetric tensor

field S of type (1,1), or (0,2), which is conservative at the critical points of E, i.e., divS = 0 at these critical points, and called it the stress-energy tensor. To study harmonic maps, P . Baird ¸ si J. Eells, 1981; A. Sanini, 1983,([1, 24]) used the tensor field S = 1 2dφ2g−φ ∗h, which satisfies divS = −τ(φ),dφ.

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The stress-energy tensor

• D. Hilbert, 1924, ([10]), associated to a functional E a symmetric tensor

field S of type (1,1), or (0,2), which is conservative at the critical points of E, i.e., divS = 0 at these critical points, and called it the stress-energy tensor. To study harmonic maps, P . Baird ¸ si J. Eells, 1981; A. Sanini, 1983,([1, 24]) used the tensor field S = 1 2dφ2g−φ ∗h, which satisfies divS = −τ(φ),dφ. φ = harmonic ⇒ divS = 0.

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The stress-energy tensor

• D. Hilbert, 1924, ([10]), associated to a functional E a symmetric tensor

field S of type (1,1), or (0,2), which is conservative at the critical points of E, i.e., divS = 0 at these critical points, and called it the stress-energy tensor. To study harmonic maps, P . Baird ¸ si J. Eells, 1981; A. Sanini, 1983,([1, 24]) used the tensor field S = 1 2dφ2g−φ ∗h, which satisfies divS = −τ(φ),dφ. φ = harmonic ⇒ divS = 0. If φ is a submersion then divS = 0 if and only if φ is harmonic.

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The stress-energy tensor

Clearly, if φ : M → N is an arbitrary Riemannian immersion (not necessarily minimal) then, as τ(φ) is normal, it follows that divS = 0.

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The stress-energy tensor

Clearly, if φ : M → N is an arbitrary Riemannian immersion (not necessarily minimal) then, as τ(φ) is normal, it follows that divS = 0. It is not interesting to study Riemannian immersions with divS = 0.

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The variational meaning of S

If φ : M → (N,h) is a fixed map, then E can be thought as a functional on the set of all Riemannian metrics on M. The critical points of this new functional are determined by S = 0.

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Biharmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume M is compact and consider The bienergy functional E2 : C∞(M,N) → R, E2 (φ) = 1 2

• M τ(φ)2 vg.

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Biharmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume M is compact and consider The bienergy functional E2 : C∞(M,N) → R, E2 (φ) = 1 2

• M τ(φ)2 vg.

The biharmonic maps are critical points of E2, i.e., for any variation {φt}t∈R of φ we have d dt

• t=0

{E2 (φt)} = 0.

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Biharmonic maps

Theorem ([11])

A smooth map φ : (Mm,g) → (Nn,h) is biharmonic if and only if the bitension field associated to φ, τ2(φ) = −∆φτ(φ)−traceg RN(dφ,τ(φ))dφ,

• vanishes. Here,

∆φ = −traceg

• ∇φ∇φ −∇φ

∇

• is the rough Laplacian on the sections of φ −1TN and

RN(X,Y)Z = ∇N

X∇N Y Z −∇N Y ∇N XZ −∇N [X,Y]Z.

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The biharmonic equation (G.Y. Jiang, 1986)

τ2(φ) = −∆φτ(φ)−traceg RN(dφ,τ(φ))dφ = 0

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The biharmonic equation (G.Y. Jiang, 1986)

τ2(φ) = −∆φτ(φ)−traceg RN(dφ,τ(φ))dφ = 0 is a fourth-order non-linear elliptic equation;

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The biharmonic equation (G.Y. Jiang, 1986)

τ2(φ) = −∆φτ(φ)−traceg RN(dφ,τ(φ))dφ = 0 is a fourth-order non-linear elliptic equation; any harmonic map is biharmonic;

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The biharmonic equation (G.Y. Jiang, 1986)

τ2(φ) = −∆φτ(φ)−traceg RN(dφ,τ(φ))dφ = 0 is a fourth-order non-linear elliptic equation; any harmonic map is biharmonic; a non-harmonic biharmonic map is called proper biharmonic;

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The stress-bienergy tensor

G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2τ(φ)2X,Y+dφ,∇τ(φ)X,Y −dφ(X),∇Yτ(φ)−dφ(Y),∇Xτ(φ). It satisfies divS2 = −τ2(φ),dφ.

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The stress-bienergy tensor

G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2τ(φ)2X,Y+dφ,∇τ(φ)X,Y −dφ(X),∇Yτ(φ)−dφ(Y),∇Xτ(φ). It satisfies divS2 = −τ2(φ),dφ. φ = biharmonic ⇒ divS2 = 0.

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The stress-bienergy tensor

G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2τ(φ)2X,Y+dφ,∇τ(φ)X,Y −dφ(X),∇Yτ(φ)−dφ(Y),∇Xτ(φ). It satisfies divS2 = −τ2(φ),dφ. φ = biharmonic ⇒ divS2 = 0. If φ is a submersion, divS2 = 0 if and only if φ is biharmonic.

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The stress-bienergy tensor

G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2τ(φ)2X,Y+dφ,∇τ(φ)X,Y −dφ(X),∇Yτ(φ)−dφ(Y),∇Xτ(φ). It satisfies divS2 = −τ2(φ),dφ. φ = biharmonic ⇒ divS2 = 0. If φ is a submersion, divS2 = 0 if and only if φ is biharmonic. If φ : M → N is a Riemannian immersion then (divS2)♯ = −τ2(φ)⊤. In general, for a Riemannian immersion, divS2 = 0.

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The variational meaning of S2

If φ : M → (N,h) is a fixed map, then E2 can be thought as a functional on the set of all Riemannian metrics on M. The critical points of this new functional are determined by S2 = 0.

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Basic facts in the submanifolds theory

Let φ : Mm → Nn be a submanifold.

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Basic facts in the submanifolds theory

Let φ : Mm → Nn be a submanifold. Locally, dφ(X) ≡ X; ∇φ

Xdφ(Y) ≡ ∇N XY;

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Basic facts in the submanifolds theory

Let φ : Mm → Nn be a submanifold. Locally, dφ(X) ≡ X; ∇φ

Xdφ(Y) ≡ ∇N XY;

Globally, φ −1(TN) =

• p∈M

Tφ(p)N; Tφ(p)N = dφp (TpM)⊕dφp (TpM)⊥ ; TM ≡

• p∈M

dφp (TpM); NM =

• p∈M

dφp (TpM)⊥ .

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Basic facts in the submanifolds theory

The Gauss equation ∇N

XY = ∇XY +B(X,Y);

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Basic facts in the submanifolds theory

The Gauss equation ∇N

XY = ∇XY +B(X,Y);

The Weingarten equation ∇N

Xη = −Aη(X)+∇⊥ X η;

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Basic facts in the submanifolds theory

The Gauss equation ∇N

XY = ∇XY +B(X,Y);

The Weingarten equation ∇N

Xη = −Aη(X)+∇⊥ X η;

B(X,Y),η = Aη(X),Y;

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Basic facts in the submanifolds theory

The Gauss equation ∇N

XY = ∇XY +B(X,Y);

The Weingarten equation ∇N

Xη = −Aη(X)+∇⊥ X η;

B(X,Y),η = Aη(X),Y; H is the mean curvature vector field H = traceB m ∈ C(NM);

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Basic facts in the submanifolds theory

The Gauss equation ∇N

XY = ∇XY +B(X,Y);

The Weingarten equation ∇N

Xη = −Aη(X)+∇⊥ X η;

B(X,Y),η = Aη(X),Y; H is the mean curvature vector field H = traceB m ∈ C(NM); If φ is a hypersurface, we denote f = traceAη, H = f

mη, f is the (m-times)

mean curvature function.

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Biconservative submanifolds; Biharmonic submanifolds

Definition

A submanifold φ : Mm → Nn is called a biharmonic submanifold if φ is a biharmonic map, i.e., τ2(φ) = 0.

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Biconservative submanifolds; Biharmonic submanifolds

Definition

A submanifold φ : Mm → Nn is called a biharmonic submanifold if φ is a biharmonic map, i.e., τ2(φ) = 0.

Definition

A submanifold φ : Mm → Nn is called a biconservative submanifold if divS2 = 0, i.e., τ2(φ)⊤ = 0.

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Mm submanifold of Nn

Mm submanifold of Nn Mm biconservative

Mm submanifold of Nn Mm biconservative Mm biharmonic

Mm submanifold of Nn Mm biconservative Mm biharmonic

Mm minimal

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Characterization results

Theorem ([5, 13, 21])

A submanifold φ : Mm → Nn is biharmonic if and only if traceA∇⊥

· H(·)+trace∇AH +trace

• RN(·,H)·

T = 0 and ∆⊥H +traceB(·,AH(·))+trace

• RN(·,H)·

⊥ = 0.

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Characterization results

Theorem ([5, 13, 21])

A submanifold φ : Mm → Nn is biharmonic if and only if traceA∇⊥

· H(·)+trace∇AH +trace

• RN(·,H)·

T = 0 and ∆⊥H +traceB(·,AH(·))+trace

• RN(·,H)·

⊥ = 0.

Proposition ([18])

Let φ : Mm → Nn be a submanifold. The following conditions are equivalent:

1

M is a biconservative submanifold;

2

traceA∇⊥

· H(·)+trace∇AH +trace

• RN(·,H)·

T = 0;

3

m 2 grad

• H2

+2traceA∇⊥

· H(·)+2trace

• RN(·,H)·

T = 0;

4

2trace∇AH − m

2 grad

• H2

= 0.

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Properties of biconservative submanifolds

Proposition

Let φ : Mm → Nn be a submanifold. If ∇AH = 0, then M is biconservative.

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Properties of biconservative submanifolds

Proposition

Let φ : Mm → Nn be a submanifold. If ∇AH = 0, then M is biconservative.

Proposition

Let φ : Mm → Nn 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.

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Properties of biconservative submanifolds

Proposition

Let φ : Mm → Nn be a submanifold. If ∇AH = 0, then M is biconservative.

Proposition

Let φ : Mm → Nn 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.

Proposition ([2])

Let φ : Mm → Nn be a submanifold. Assume that M is pseudoumbilical, i.e., AH = H2I, and m = 4. Then M is CMC.

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Characterization theorems

Theorem([2, 22])

If φ : Mm → Nm+1 is a hypersurface, then M is biharmonic if and only if 2A(gradf)+f gradf −2f

• RicciN(η)

T = 0, and ∆f +f|A|2 −f RicciN(η,η) = 0, where η is the unit normal vector field along M in N.

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Characterization theorems

Theorem([2, 22])

If φ : Mm → Nm+1 is a hypersurface, then M is biharmonic if and only if 2A(gradf)+f gradf −2f

• RicciN(η)

T = 0, and ∆f +f|A|2 −f RicciN(η,η) = 0, where η is the unit normal vector field along M in N. A hypersurface φ : Mm → Nm+1(c) is biconservative if and only if A(gradf) = − f 2 gradf.

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Characterization theorems

Theorem([2, 22])

If φ : Mm → Nm+1 is a hypersurface, then M is biharmonic if and only if 2A(gradf)+f gradf −2f

• RicciN(η)

T = 0, and ∆f +f|A|2 −f RicciN(η,η) = 0, where η is the unit normal vector field along M in N. A hypersurface φ : Mm → Nm+1(c) is biconservative if and only if A(gradf) = − f 2 gradf. Every CMC hypersurface in Nm+1(c) is biconservative.

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Biconservative surfaces

Let φ : M2 → N3(c) be a non-CMC biconservative surface. Local proprieties Global properties

Biconservative surfaces

Let φ : M2 → N3(c) be a non-CMC biconservative surface. Local proprieties Global properties f > 0 and gradf = 0

• n M

Biconservative surfaces

Let φ : M2 → N3(c) be a non-CMC biconservative surface. Local proprieties Global properties f > 0 and gradf = 0

• n M

f > 0 on M and gradf = 0

• n an open and dense

subset of M

Biconservative surfaces

Let φ : M2 → N3(c) be a non-CMC biconservative surface. Local proprieties Global properties f > 0 and gradf = 0

• n M

f > 0 on M and gradf = 0

• n an open and dense

subset of M extrinsic intrinsic

Biconservative surfaces

Let φ : M2 → N3(c) be a non-CMC biconservative surface. Local proprieties Global properties f > 0 and gradf = 0

• n M

f > 0 on M and gradf = 0

• n an open and dense

subset of M extrinsic intrinsic extrinsic intrinsic

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Local extrinsic properties

Theorem ([3])

Let φ : M2 → N3(c) be a biconservative surface with f > 0 and gradf = 0 at any point in M. Then we have f∆f +|gradf|2 + 4 3cf 2 −f 4 = 0, (1) where ∆ is the Laplace-Beltrami operator on M.

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Local extrinsic properties

Theorem ([3])

Let φ : M2 → N3(c) be a biconservative surface with f > 0 and gradf = 0 at any point in M. Then we have f∆f +|gradf|2 + 4 3cf 2 −f 4 = 0, (1) where ∆ is the Laplace-Beltrami operator on M. In fact, we proved that on a neighborhood of any point in M, there exists a local chart (U;u,v) such that f = f(u,v) = f(u) and (1) is equivalent with ff ′′ − 7 4

• f ′2 − 4

3cf 2 +f 4 = 0, (2) i.e., f has to satisfy a second order ODE.

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Local intrinsic properties

Using the Gauss equation, K = c+detA, we get f 2 = 4 3(c−K). (3)

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Local intrinsic properties

Using the Gauss equation, K = c+detA, we get f 2 = 4 3(c−K). (3)

Theorem

Let φ : M2 → N3(c) be a biconservative surface with f > 0 and gradf = 0 at any point of M. Then we obtain (c−K)∆K −|gradK|2 − 8 3K(c−K)2 = 0, (4) where ∆ is Laplace-Beltrami operator on M.

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The intrinsic problem

We want to determine the necessary and sufficient conditions such that an abstract surface

• M2,g
• to admit, locally, a biconservative embedding in N3(c)

with f > 0 and gradf = 0.

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Ricci problem

Given an abstract surface

• M2,g
• , we want to determine the necessary

and sufficient conditions such that it admits, locally, a minimal embedding in N3(c).

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Ricci problem

Given an abstract surface

• M2,g
• , we want to determine the necessary

and sufficient conditions such that it admits, locally, a minimal embedding in N3(c). It was proved (see [16, 23]) that if

• M2,g
• is an abstract surface such that

c−K > 0 on M, where c ∈ R is a constant, then, locally, it admits a minimal embedding in N3(c) if and only if (c−K)∆K −|gradK|2 −4K(c−K)2 = 0. (5)

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Ricci problem

Given an abstract surface

• M2,g
• , we want to determine the necessary

and sufficient conditions such that it admits, locally, a minimal embedding in N3(c). It was proved (see [16, 23]) that if

• M2,g
• is an abstract surface such that

c−K > 0 on M, where c ∈ R is a constant, then, locally, it admits a minimal embedding in N3(c) if and only if (c−K)∆K −|gradK|2 −4K(c−K)2 = 0. (5) Condition (5) is called the Ricci condition with respect to c, or simple the Ricci condition. If (5) holds, then, locally, M admits a one-parametric family of minimal embeddings in N3(c).

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The link between the biconservativity and the Ricci condition with respect to c = 0

We can notice that relations (4) and (5) are very similar. In [7], we study the relationship between them.

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The link between the biconservativity and the Ricci condition with respect to c = 0

We can notice that relations (4) and (5) are very similar. In [7], we study the relationship between them.

• M2,g
• bicons. in R3
• M2,g
• satisfies (4), K < 0

The link between the biconservativity and the Ricci condition with respect to c = 0

We can notice that relations (4) and (5) are very similar. In [7], we study the relationship between them.

• M2,g
• bicons. in R3
• M2,g
• satisfies (4), K < 0
• M2,g
• satisfies (4), K < 0
• M2,g1/2 = √−Kg
• Ricci, K1/2 < 0

The link between the biconservativity and the Ricci condition with respect to c = 0

We can notice that relations (4) and (5) are very similar. In [7], we study the relationship between them.

• M2,g
• bicons. in R3
• M2,g
• satisfies (4), K < 0
• M2,g
• satisfies (4), K < 0
• M2,g1/2 = √−Kg
• Ricci, K1/2 < 0
• M2,g
• Ricci, K < 0
• M2,g−1 = (−K)−1 g
• satisfies (4), K−1 < 0

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The link between the biconservativity and the Ricci condition with respect to c

Theorem ([7])

Let

• M2,g
• be a biconservative surface in N3(c). If f > 0 and gradf = 0 on M,

then on an open and dense subset of M,

• M2,(c−K)rg
• is a Ricci surface,

where r is a function which locally satisfies K +∆ 1 4 log(c−Kr)+ r 2 log(c−K)

• = 0,

and Kr, which denotes the Gaussian curvature on M corresponding to the metric (c−K)rg, is given by Kr = (c−K)−r 3−4r 3 K + 1 2 log(c−K)∆r +(c−K)−1g(gradr,gradK)

• .

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Local intrinsic characterization of biconservative surfaces in N3(c)

Theorem ([7])

Let

• M2,g
• be an abstract surface and c ∈ R a constant. Then, locally, M can

be isometrically embedded in N3(c) as a biconservative surface with f > 0 and 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) .

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Local intrinsic characterization of biconservative surfaces in N3(c)

Theorem ([7])

Let

• M2,g
• be an abstract surface and c ∈ R a constant. Then, locally, M can

be isometrically embedded in N3(c) as a biconservative surface with f > 0 and 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) .

Corollary

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.

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Local intrinsic characterization

Theorem ([19])

Let

• M2,g
• be an abstract surface with c−K(p) > 0 and (gradK)(p) = 0 at any point

p ∈ M, where c ∈ R is a constant. Let X1 = (gradK)/|gradK| and X2 ∈ C(TM) be two vector fields on M such that {X1(p),X2(p)} is a positively oriented basis at any point of p ∈ M. Then, the following conditions are equivalent: (a) the level curves of K are circles in M with constant curvature κ = 3|gradK| 8(c−K) = 3X1K 8(c−K);

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Local intrinsic characterization

Theorem ([19])

Let

• M2,g
• be an abstract surface with c−K(p) > 0 and (gradK)(p) = 0 at any point

p ∈ M, where c ∈ R is a constant. Let X1 = (gradK)/|gradK| and X2 ∈ C(TM) be two vector fields on M such that {X1(p),X2(p)} is a positively oriented basis at any point of p ∈ M. Then, the following conditions are equivalent: (a) the level curves of K are circles in M with constant curvature κ = 3|gradK| 8(c−K) = 3X1K 8(c−K); (b) X2 (X1K) = 0 and ∇X2X2 = −3X1K 8(c−K)X1;

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Local intrinsic characterization

Theorem ([19])

Let

• M2,g
• be an abstract surface with c−K(p) > 0 and (gradK)(p) = 0 at any point

p ∈ M, where c ∈ R is a constant. Let X1 = (gradK)/|gradK| and X2 ∈ C(TM) be two vector fields on M such that {X1(p),X2(p)} is a positively oriented basis at any point of p ∈ M. Then, the following conditions are equivalent: (a) the level curves of K are circles in M with constant curvature κ = 3|gradK| 8(c−K) = 3X1K 8(c−K); (b) X2 (X1K) = 0 and ∇X2X2 = −3X1K 8(c−K)X1; (c) locally, the metric g can be written as g = (c−K)−3/4 du2 +dv2 , where (u,v) are local coordinates positively oriented, K = K(u), and K′ > 0;

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Local intrinsic characterization

(d) locally, the metric g can be written as g = e2ϕ du2 +dv2 , where (u,v) are local coordinates positively

• riented, and ϕ = ϕ(u) satisfies the equation

ϕ′′ = e−2ϕ/3 −ce2ϕ (6) and the condition ϕ′ > 0; moreover, the solutions of the above equation, u = u(ϕ), are u =

ϕ

ϕ0

dτ √ −3e−2τ/3 −ce2τ +a +u0, where ϕ is in some open interval I and a,u0 ∈ R are constants;

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Local intrinsic characterization

(d) locally, the metric g can be written as g = e2ϕ du2 +dv2 , where (u,v) are local coordinates positively

• riented, and ϕ = ϕ(u) satisfies the equation

ϕ′′ = e−2ϕ/3 −ce2ϕ (6) and the condition ϕ′ > 0; moreover, the solutions of the above equation, u = u(ϕ), are u =

ϕ

ϕ0

dτ √ −3e−2τ/3 −ce2τ +a +u0, where ϕ is in some open interval I and a,u0 ∈ R are constants; (e) locally, the metric g can be written as g = e2ϕ du2 +dv2 , where (u,v) are local coordinates positively

• riented, and ϕ = ϕ(u) satisfies the equation

3ϕ′′′ +2ϕ′ϕ′′ +8ce2ϕϕ′ = 0 (7) and the conditions ϕ′ > 0 and c+e−2ϕϕ′′ > 0; moreover, the solutions of the above equation, u = u(ϕ), are u =

ϕ

ϕ0

dτ √ −3be−2τ/3 −ce2τ +a +u0, where ϕ is in some open interval I and a,b,u0 ∈ R are constants, b > 0.

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Local intrinsic properties, c = 0

Proposition ([17])

The solutions of the equation 3ϕ′′′ +2ϕ′ϕ′′ = 0 which satisfy the conditions ϕ′ > 0 and ϕ′′ > 0, up to affine transformations of the parameter with α > 0 (u = α ˜ u+β), are given by ϕ(u) = 3log(coshu)+constant, u > 0.

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Local intrinsic properties

Remarks

If c = 0, we have a one-parameter family of solutions of equation (7), i.e., gC0 = C0(coshu)6 du2 +dv2 , C0 > 0.

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Local intrinsic properties

Remarks

If c = 0, we have a one-parameter family of solutions of equation (7), i.e., gC0 = C0(coshu)6 du2 +dv2 , C0 > 0. If c = 0, then we cannot determine explicitly ϕ = ϕ(u), but we have u = u(ϕ). Another way to see that we have only a one-parameter family of solutions of equation (7) is to rewrite the metric g in certain non-isothermal coordinates.

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Local intrinsic properties

Remarks

If c = 0, we have a one-parameter family of solutions of equation (7), i.e., gC0 = C0(coshu)6 du2 +dv2 , C0 > 0. If c = 0, then we cannot determine explicitly ϕ = ϕ(u), but we have u = u(ϕ). Another way to see that we have only a one-parameter family of solutions of equation (7) is to rewrite the metric g in certain non-isothermal coordinates. Further, we consider only the c = 1 case.

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Local intrinsic properties, c = 1

Proposition ([17])

Let

• M2,g
• be an abstract surface with g = e2ϕ(u)(du2 +dv2), where u = u(ϕ)

satisfies u =

ϕ

ϕ0

dτ

• −3be−2τ/3 −e2τ +a

+u0, where ϕ is in some open interval I, a,b ∈ R are positive constants, and u0 ∈ R is a constant. Then

• M2,g
• is isometric to
• DC1,gC1 =

3 ξ 2 −ξ 8/3 +3C1ξ 2 −3 dξ 2 + 1 ξ 2 dθ 2

• ,

where DC1 = (ξ01,ξ02)×R, C1 ∈

• 4/
• 33/2

,∞

• is a positive constant, and ξ01

and ξ02 are the positive vanishing points of −ξ 8/3 +3C1ξ 2 −3, with 0 < ξ01 < ξ02.

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Local intrinsic properties, c = 1

Remark

Let us consider

• DC1,gC1 =

3 ξ 2 −ξ 8/3 +3C1ξ 2 −3 dξ 2 + 1 ξ 2 dθ 2

• and

 DC′

1,gC′ 1 =

3 ˜ ξ 2

• − ˜

ξ 8/3 +3C′

1 ˜

ξ 2 −3 d ˜ ξ 2 + 1 ˜ ξ 2 d ˜ θ 2  . The surfaces

• DC1,gC1
• and
• DC′

1,gC′ 1

• are isometric if and only if C1 = C′

1 and

the isometry is Θ(ξ,θ) = (ξ,±θ +constant). Therefore, we have a

• ne-parameter family of surfaces.

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Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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Complete biconservative surfaces in R3

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In this section we consider the global problem and,

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In this section we consider the global problem and, from extrinsic point of view, we construct biconservative surfaces in R3 with f > 0 at any point of the surface and gradf = 0 at any point of an

• pen and dense subset.

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In this section we consider the global problem and, from extrinsic point of view, we construct biconservative surfaces in R3 with f > 0 at any point of the surface and gradf = 0 at any point of an

• pen and dense subset.

from intrinsic point of view, we construct a complete abstract surface

• M2,g
• with K < 0 on M and gradK = 0 on an open and dense subset of

M, which admits a biconservative immersion in R3, defined on whole the surface M, with f > 0 on M and gradf = 0 on that open and dense subset

• f M.

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S ˜

C0

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S ˜

C0

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S ˜

C0

˜ S ˜

C0

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Local extrinsic result

Theorem ([9])

Let M2 be a surface in R3 with f(p) > 0 and (gradf)(p) = 0 at any p ∈ M. Then, M2 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 −4κ4.

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Local extrinsic result

Theorem ([3])

Let M2 be a biconservative surface in R3 with f(p) > 0 and (gradf)(p) = 0 at any p ∈ M. Then, locally, the surface can be parametrized by X ˜

C0(ρ,v) =

• ρ cosv,ρ sinv,u ˜

C0(ρ)

• ,

where u ˜

C0(ρ) =

3 2 ˜ C0

• ρ1/3
• ˜

C0ρ2/3 −1+ 1 ˜ C0 log

• ˜

C0ρ1/3 +

• ˜

C0ρ2/3 −1

• with ˜

C0 a positive constant and ρ ∈

• ˜

C−3/2 ,∞

• .

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Any two such surfaces are not locally isometric, so we have a

• ne-parameter family of biconservative surfaces in R3.

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Any two such surfaces are not locally isometric, so we have a

• ne-parameter family of biconservative surfaces in R3.

These surfaces are NOT complete.

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Any two such surfaces are not locally isometric, so we have a

• ne-parameter family of biconservative surfaces in R3.

These surfaces are NOT complete. We denote by S ˜

C0 the image X ˜ C0

• ˜

C−3/2 ,∞

• ×R
• . The boundary of S ˜

C0,

i.e., S ˜

C0 \S ˜ C0, is the circle

• ˜

C−3/2 cosv, ˜ C−3/2 sinv,0

• , which lies in the xOy
• plane. At a boundary point, the tangent plane to the closure S ˜

C0 of S ˜ C0 is

parallel to Oz. Moreover, along the boundary, the mean curvature function is constant f ˜

C0 =

• 2 ˜

C3/2

• /3 and gradf ˜

C0 = 0.

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Any two such surfaces are not locally isometric, so we have a

• ne-parameter family of biconservative surfaces in R3.

These surfaces are NOT complete. We denote by S ˜

C0 the image X ˜ C0

• ˜

C−3/2 ,∞

• ×R
• . The boundary of S ˜

C0,

i.e., S ˜

C0 \S ˜ C0, is the circle

• ˜

C−3/2 cosv, ˜ C−3/2 sinv,0

• , which lies in the xOy
• plane. At a boundary point, the tangent plane to the closure S ˜

C0 of S ˜ C0 is

parallel to Oz. Moreover, along the boundary, the mean curvature function is constant f ˜

C0 =

• 2 ˜

C3/2

• /3 and gradf ˜

C0 = 0.

Thus, in order to obtain a complete biconservative surface in R3, we proved that we can glue two biconservative surfaces S ˜

C0 and S ˜ C′

0, at the

level of C∞ smoothness, only along the boundary and, in this case, ˜ C0 = ˜ C′

0.

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Global extrinsic result

Proposition ([15, 17])

If we consider the symmetry of GrafuC, with respect to the Oρ(= Ox) axis, we get a smooth, complete, biconservative surface ˜ S ˜

C0 in R3. Moreover, its mean

curvature function ˜ f ˜

C0 is positive and grad ˜

f ˜

C0 is different from zero at any point

• f an open dense subset of ˜

S ˜

C0.

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Uniqueness

Theorem ([20])

Let M2 be a biconservative regular surface in R3. If M is compact, then M is CMC.

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Uniqueness

Theorem ([20])

Let M2 be a biconservative regular surface in R3. If M is compact, then M is CMC.

Theorem

Let M2 be a biconservative regular surface in R3. Assume that M is a complete, non-compact surface and the number of the connected components of W = {p ∈ M | (gradf)(p) = 0} is finite. Then M = ˜ S ˜

C0.

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Further, we change the point of view and use the intrinsic characterization of biconservative surfaces in R3.

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Global intrinsic result

Theorem ([17])

Let

• R2,gC0 = C0 (coshu)6

du2 +dv2 be a surface, where C0 ∈ R is a positive constant. Then we have:

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Global intrinsic result

Theorem ([17])

Let

• R2,gC0 = C0 (coshu)6

du2 +dv2 be a surface, where C0 ∈ R is a positive constant. Then we have: (a) the metric on R2 is complete;

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Global intrinsic result

Theorem ([17])

Let

• R2,gC0 = C0 (coshu)6

du2 +dv2 be a surface, where C0 ∈ R is a positive constant. Then we have: (a) the metric on R2 is complete; (b) the Gaussian curvature is given by KC0(u,v) = KC0(u) = − 3 C0 (coshu)8 < 0, K′

C0(u) =

24sinhu C0 (coshu)9 , and therefore gradKC0 = 0 at any point of R2 \Ov; (c) the immersion ϕC0 :

• R2,gC0
• → R3 given by

ϕC0(u,v) =

• σ 1

C0(u)cos(3v),σ 1 C0(u)sin(3v),σ 2 C0(u)

• is biconservative in R3, where

σ 1

C0(u) =

√C0 3 (coshu)3 , σ 2

C0(u) =

√C0 2 1 2 sinh(2u)+u

• ,

u ∈ R.

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Theorem ([19])

Let

• R2,gC0
• . Then
• R2,−KC0gC0
• satisfies the Ricci condition and can be

minimal immersed in R3 as a helicoid or a catenoid.

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Theorem ([19])

Let

• R2,gC0
• . Then
• R2,−KC0gC0
• satisfies the Ricci condition and can be

minimal immersed in R3 as a helicoid or a catenoid.

Proposition

Let

• R2,gC0
• . Then
• R2,−KC0gC0
• has constant Gaussian curvature 1/3 and it

is not complete. Moreover,

• R2,−KC0gC0
• is the universal cover of the surface
• f revolution in R3 given by

Z(u,v) =

• α(u)cosh

√ 3 a v

• ,α(u)sinh

√ 3 a v

• ,β(u)
• ,

(u,v) ∈ R2, where a ∈ (0, √ 3] and α(u) = a coshu, β(u) =

u

• (3−a2)cosh2 τ +a2

cosh2 τ dτ.

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Remark

When a = √ 3, the immersion Z has only umbilical points and the image Z

• R2

is the round sphere of radius √ 3, without the North and the South poles. Moreover, if a ∈ (0, √ 3), then Z has no umbilical points.

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Remark

When a = √ 3, the immersion Z has only umbilical points and the image Z

• R2

is the round sphere of radius √ 3, without the North and the South poles. Moreover, if a ∈ (0, √ 3), then Z has no umbilical points. Concerning the biharmonic surfaces in R3 we have the following non-existence result.

Theorem ([4, 6])

There exists no proper biharmonic surface in R3.

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Complete biconservative surfaces in S3

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In this section we consider the global problem and construct biconservative surfaces in S3 with f > 0 at any point of the surface and gradf = 0 at any point

• f an open and dense subset.

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• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY

• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY φC1 = φ± C1,c1 BICONSERVATIVE

S3

• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY φC1 = φ± C1,c1 BICONSERVATIVE

S3

ψC1,C∗ 1 = ψ± C1,C∗ 1,c∗ 1 ISOMETRY

S±

C1,C∗

1,c∗ 1 ⊂ R3

• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY φC1 = φ± C1,c1 BICONSERVATIVE

S3

ψC1,C∗ 1 = ψ± C1,C∗ 1,c∗ 1 ISOMETRY

S±

C1,C∗

1,c∗ 1 ⊂ R3

˜ SC1,C∗

1 ⊂ R3 complete p l a y i n g w i t h t h e c

• n

s t . c ∗ 1 a n d ±

• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY φC1 = φ± C1,c1 BICONSERVATIVE

S3

ψC1,C∗ 1 = ψ± C1,C∗ 1,c∗ 1 ISOMETRY

S±

C1,C∗

1,c∗ 1 ⊂ R3

˜ SC1,C∗

1 ⊂ R3 complete p l a y i n g w i t h t h e c

• n

s t . c ∗ 1 a n d ±

playing with the const. c1 and ±, ΦC1,C∗

1 63 / 88

The projection of ΦC1,C∗

1 on the Ox1x2 plane is a curve which lies in the

annulus of radii

• 1−1/
• C1ξ 2

01

• and
• 1−1/
• C1ξ 2

02

• . It has self-intersections

and is dense in the annulus. Choosing C1 = C∗

1 = 1, we obtain

x1 x2

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The signed curvature of the profile curve of ˜ SC1,C∗

1.

ν κ

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The signed curvature of the curve obtained projecting Φ1,1 on the Ox1x2 plane. ν κ

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Local extrinsic result

Theorem ([3])

Let M2 be a biconservative surface in S3 with f(p) > 0 and (gradf)(p) = 0 for any p ∈ M. Then, locally, the surface viewed in R4, can be parametrized by Y ˜

C1(u,v) = σ(u)+ 4κ(u)−3/4

3 ˜ C1

• f 1(cosv−1)+f 2 sinv
• ,

where ˜ C1 ∈

• 64/
• 35/4

,∞

• is a positive constant; f 1,f 2 ∈ R4 are two constant orthonormal vectors; σ(u) is a

curve parametrized by arclength that satisfies σ(u),f 1 = 4κ(u)−3/4 3 ˜ C1 , σ(u),f 2 = 0, and, as a curve in S2, its curvature κ = κ(u) is a positive non-constant solution of the following ODE κ′′κ = 7 4 (κ′)2 + 4 3 κ2 −4κ4 such that (κ′)2 = − 16 9 κ2 −16κ4 + ˜ C1κ7/2.

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In the above theorem if we consider f 1 = e3 and f 2 = e4 and change the coordinates (u,v) in (κ,v), we obtain

Y ˜

C1(κ,v) =

• 1−
• 4

3√ ˜ C1 κ−3/4

2 cosµ(κ),

• 1−
• 4

3√ ˜ C1 κ−3/4

2 sinµ(κ),

4 3√ ˜ C1 κ−3/4 cosv, 4 3√ ˜ C1 κ−3/4 sinv

• ,

(8)

where (κ,v) ∈ (κ01,κ02)×R, κ01 and κ02 are positive solutions of the equation −16 9 κ2 −16κ4 + ˜ C1κ7/2 = 0 and µ(κ) = ±

κ

κ0 E(τ) dτ +c0, with c0 ∈ R and κ0 ∈ (κ01,κ02).

We choose κ0 = (3 ˜ C1/64)2. An alternative expression for Y ˜

C1 was given in [8].

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Further, as in the R3 case, we change the point of view and we use the intrinsic characterization of biconservative surfaces in S3.

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Further, as in the R3 case, we change the point of view and we use the intrinsic characterization of biconservative surfaces in S3. The surface

• DC1,gC1
• defined in the previous subsection is NOT complete but

it has the following properties.

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Theorem ([17])

Let (DC1,gC1). Then

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Theorem ([17])

Let (DC1,gC1). Then (a) KC1(ξ,θ) = K(ξ,θ), 1−K(ξ,θ) = 1 9ξ 8/3 > 0, K′(ξ) = − 8 27ξ 5/3 and gradK = 0 at any point in DC1;

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Theorem ([17])

Let (DC1,gC1). Then (a) KC1(ξ,θ) = K(ξ,θ), 1−K(ξ,θ) = 1 9ξ 8/3 > 0, K′(ξ) = − 8 27ξ 5/3 and gradK = 0 at any point in DC1; (b) the immersion φC1 : (DC1,gC1) → S3 given by

φC1(ξ,θ) =

• 1−

1 C1ξ 2 cosζ(ξ),

• 1−

1 C1ξ 2 sinζ(ξ), cos(√C1θ) √C1ξ , sin(√C1θ) √C1ξ

• ,

is biconservative in S3, where ζ(ξ) = ±

ξ

ξ00 E(τ) dτ +c1, with c1 ∈ R ¸

si ξ00 ∈ (ξ01,ξ02).

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Theorem ([17])

Let (DC1,gC1). Then (a) KC1(ξ,θ) = K(ξ,θ), 1−K(ξ,θ) = 1 9ξ 8/3 > 0, K′(ξ) = − 8 27ξ 5/3 and gradK = 0 at any point in DC1; (b) the immersion φC1 : (DC1,gC1) → S3 given by

φC1(ξ,θ) =

• 1−

1 C1ξ 2 cosζ(ξ),

• 1−

1 C1ξ 2 sinζ(ξ), cos(√C1θ) √C1ξ , sin(√C1θ) √C1ξ

• ,

is biconservative in S3, where ζ(ξ) = ±

ξ

ξ00 E(τ) dτ +c1, with c1 ∈ R ¸

si ξ00 ∈ (ξ01,ξ02). Alegem ξ00 = (9C1/4)3/2.

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The key ingredient

The key ingredient in the construction of the complete biconservative surface in S3 is to notice that

• DC1,gC1
• is locally and intrinsically isometric with a

surface of revolution in R3. Then, we construct a complete surface of revolution in R3, which on an open and dense subset is locally isometric with

• DC1,gC1
• .

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Theorem

Let

• DC1,gC1
• . Then
• DC1,gC1
• is the universal cover of the surface of

revolution in R3 given by ψC1,C∗

1(ξ,θ) =

• χ(ξ)cos θ

C∗

1

,χ(ξ)sin θ C∗

1

,ν(ξ)

• ,

(9) where χ(ξ) = C∗

1/ξ,ν(ξ) = ±

ξ

ξ00 E(τ) dτ +c∗ 1, C∗ 1 ∈

• 0,
• C1 −4/33/2−1/2

is a positive constant and c∗

1 ∈ R.

72 / 88

Content

1

The motivation of the research topic General context Harmonic maps Biharmonic maps

2

Properties of biconservative submanifolds Biconservative submanifolds – Biharmonic submanifolds Biconservative surfaces – Ricci surfaces Local intrinsic characterization of biconservative surfaces in N3(c) Complete biconservative surfaces Biconservative surfaces in Nn

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We recall that a biconservative surface is characterized by divS2 = 0. So, first, we give some properties of a symmetric tensor field T of type (1,1) that satisfies divT = 0 and then we focus on the biconservative case.

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Properties of T

Theorem ([18])

Let

• M2,g
• be a surface and let T a symmetric tensor field of type (1,1). Then,

any two relations involve any of the others

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Properties of T

Theorem ([18])

Let

• M2,g
• be a surface and let T a symmetric tensor field of type (1,1). Then,

any two relations involve any of the others

1

divT = 0;

75 / 88

Properties of T

Theorem ([18])

Let

• M2,g
• be a surface and let T a symmetric tensor field of type (1,1). Then,

any two relations involve any of the others

1

divT = 0;

2

t is constant, where t = traceT;

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Properties of T

Theorem ([18])

Let

• M2,g
• be a surface and let T a symmetric tensor field of type (1,1). Then,

any two relations involve any of the others

1

divT = 0;

2

t is constant, where t = traceT;

3

T (∂z),∂z is holomorphic;

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Properties of T

Theorem ([18])

Let

• M2,g
• be a surface and let T a symmetric tensor field of type (1,1). Then,

any two relations involve any of the others

1

divT = 0;

2

t is constant, where t = traceT;

3

T (∂z),∂z is holomorphic;

4

T is a Codazzi tensor field.

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T = S2

M2 bicons. |H| const. S2 (∂z),∂z holomorphic S2 Codazzi

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T = S2

M2 bicons. |H| const. S2 (∂z),∂z holomorphic S2 Codazzi

T = AH

divAH = 0 |H| const. AH (∂z),∂z holomorphic AH Codazzi

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We recall that for a surface S2 = −2|H|2I +4AH, therefore, in general, S2 is Codazzi ⇔ AH is Codazzi divS2 = 0 ⇔ divAH = 0. divS2 = −2grad

• |H|2

+4divAH.

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

2

|H| is constant;

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

2

|H| is constant;

3

AH (∂z),∂z is holomorphic;

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

2

|H| is constant;

3

AH (∂z),∂z is holomorphic;

4

AH is a Codazzi tensor field.

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

2

|H| is constant;

3

AH (∂z),∂z is holomorphic;

4

AH is a Codazzi tensor field.

Corollary

Let ϕ : M2 → N3(c) be a CMC surface. Then

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

2

|H| is constant;

3

AH (∂z),∂z is holomorphic;

4

AH is a Codazzi tensor field.

Corollary

Let ϕ : M2 → N3(c) be a CMC surface. Then

1

M is a biconservative surface;

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

2

|H| is constant;

3

AH (∂z),∂z is holomorphic;

4

AH is a Codazzi tensor field.

Corollary

Let ϕ : M2 → N3(c) be a CMC surface. Then

1

M is a biconservative surface;

2

AH is a Codazzi tensor field;

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Properties of biconservative surfaces

Theorem

Let ϕ : M2 → Nn be a surface. Then, any two relations involve any of the others

1

M is a biconservative surface;

2

|H| is constant;

3

AH (∂z),∂z is holomorphic;

4

AH is a Codazzi tensor field.

Corollary

Let ϕ : M2 → N3(c) be a CMC surface. Then

1

M is a biconservative surface;

2

AH is a Codazzi tensor field;

3

AH (∂z),∂z is holomorphic.

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Rough Laplacian ∆RS2

Proposition

Let

• M2,g
• be a surface and let T be a symmetric tensor field of type (1,1).

Assume that divT = 0. Then

∆RT = −2KT +tKI +(∆t)I +∇gradt, (10)

where ∆RT = −trace

• ∇2T
• , t = traceT and I is the identity tensor field of type

(1,1).

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Rough Laplacian ∆RS2

Proposition

Let

• M2,g
• be a surface and let T be a symmetric tensor field of type (1,1).

Assume that divT = 0. Then

∆RT = −2KT +tKI +(∆t)I +∇gradt, (10)

where ∆RT = −trace

• ∇2T
• , t = traceT and I is the identity tensor field of type

(1,1).

Corollary

If ϕ : M2 → Nn is a biconservative surface, then

∆RS2 = −2KS2 +∇grad

• |τ(ϕ)|2

+

• K|τ(ϕ)|2 +∆|τ(ϕ)|2

I, (11)

where K is the Gaussian curvature of M.

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Rough Laplacian ∆RS2

Proposition

Let

• M2,g
• be a surface and let T be a symmetric tensor field of type (1,1).

Assume that divT = 0. Then

∆RT = −2KT +tKI +(∆t)I +∇gradt, (10)

where ∆RT = −trace

• ∇2T
• , t = traceT and I is the identity tensor field of type

(1,1).

Corollary

If ϕ : M2 → Nn is a biconservative surface, then

∆RS2 = −2KS2 +∇grad

• |τ(ϕ)|2

+

• K|τ(ϕ)|2 +∆|τ(ϕ)|2

I, (11)

where K is the Gaussian curvature of M.

Formula (11) was obtained in [14] but for biharmonic maps (a stronger hypothesis) from surfaces and in a different way.

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The Simons type formula for S2

Proposition

Let ϕ : M2 → Nn be a biconservative surface. Then,

1 2∆|S2|2 =

−2K |S2|2 +div

• S2,grad
• |τ(ϕ)|2
• ♯

+K|τ(ϕ)|4 + 1

2∆

• |τ(ϕ)|4

+

• grad
• |τ(ϕ)|2

2 −|∇S2|2 . (12)

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Proposition

Let ϕ : M2 → Nn be a biconservative surface and assume that M is compact. Then

• M
• |∇S2|2 +2K
• |S2|2 − |τ(ϕ)|4

2

• vg =
• M
• grad
• |τ(ϕ)|2
• 2

vg,

• r, equivalent,
• M
• |∇AH|2 +2K
• |AH|2 −2|H|4

vg = 5 2

• M
• grad
• |H|2
• 2

vg.

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Properties of biconservative surfaces in Nn

M2 surface, ∇AH = 0

M2 bicons.

Theorem

Let ϕ : M2 → Nn be a CMC biconservative surface and assume that M is

• compact. If K ≥ 0, then ∇AH = 0 and M is flat or M is pseudoumbilical.

Properties of biconservative surfaces in Nn

M2 surface, ∇AH = 0

M2 bicons. M2 is CMC, compact, K ≥ 0

Theorem

Let ϕ : M2 → Nn be a CMC biconservative surface and assume that M is

• compact. If K ≥ 0, then ∇AH = 0 and M is flat or M is pseudoumbilical.

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References I

[1] P . Baird, J. Eells, A conservation law for harmonic maps, Geometry Symposium Utrecht 1980, 1–25, Lecture Notes in Math. 894, Springer, Berlin-New York, 1981. [2]

• A. Balmu¸

s, S. Montaldo, C. Oniciuc, Biharmonic PNMC submanifolds in spheres, Ark. Mat. 51 (2013), 197–221. [3]

• 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. [4] B-Y. Chen, Some open problems and conjectures on submanifolds of finte type, Soochow I. Math. 17 (1991), 169–188. [5] B-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1. World Scientific Publishing Co., Singapore, 1984.

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References II

[6] B-Y. Chen, S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A. 45 (1991), 323–347. [7]

• D. Fetcu, S. Nistor, C. Oniciuc,

On biconservative surfaces in 3-dimensional space forms, Comm. Anal.

• Geom. (5) 24 (2016), 1027–1045.

[8]

• Y. Fu,

Explicit classification of biconservative surfaces in Lorentz 3-space forms, Ann. Mat. Pura Appl.(4) 194 (2015), 805–822. [9]

• Th. Hasanis, Th. Vlachos,

Hypersurfaces in E4 with harmonic mean curvature vector field, Math.

• Nachr. 172 (1995), 145–169.

[10] D. Hilbert, Die grundlagen der physik, Math. Ann. 92 (1924), 1–32.

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References III

[11] G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A7(4) (1986), 389–402. [12] G. Y. Jiang, The conservation law for 2-harmonic maps between Riemannian manifolds, Acta Math. Sinica 30 (1987), 220–225. [13] E. Loubeau, S. Montaldo, C. Oniciuc, The stress-energy tensor for biharmonic maps, Math. Z. 259 (2008), 503–524. [14] E. Loubeau, C. Oniciuc, Biharmonic surfaces of constant mean curvature, Pacific J. Math. 271 (2014), 213–230. [15] S. Montaldo, C. Oniciuc, A. Ratto, Proper biconservative immersions into the Euclidean space, Ann. Mat. Pura Appl. (4) 195 (2016), 403–422.

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References IV

[16] A. Moroianu, S. Moroianu, Ricci surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5) XIV (2015), 1093–1118. [17] S. Nistor, Complete biconservative surfaces in R3 and S3, J. Geom. Phys. 110 (2016) 130-153. [18] S. Nistor-Barna, On biconservative surfaces, work in progress. [19] S. Nistor-Barna, C. Oniciuc Global properties of biconservative surfaces in R3 and S3, accepted. [20] S. Nistor-Barna, C. Oniciuc On the uniqueness of complete biconservative surfaces in R3 , work in progress.

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References V

[21] C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. Stiint. Univ. Al.I. Cuza Iasi Mat (N.S.) 48 (2002), 237–248. [22] Y.-L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math. 248 (2010), 217–232. [23] G. Ricci-Curbastro, Sulla teoria intrinseca delle superficie ed in ispecie di quelle di 2◦ grado,

• Ven. Ist. Atti (7) VI (1895), 445–488.

[24] A. Sanini, Applicazioni tra varietà riemanniane con energia critica rispetto a deformazioni di metriche, Rend. Mat. 3 (1983), 53–63.

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Thank you!

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