Biconservative surfaces in Riemannian manifolds Simona Nistor - - PowerPoint PPT Presentation

Biconservative surfaces in Riemannian manifolds

Simona Nistor

“Alexandru Ioan Cuza” University of Ia¸ si

Harmonic Maps Workshop Brest, May 15-18, 2017

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Content

1

The motivation of the research topic

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Content

1

The motivation of the research topic

2

Introducing the biconservative immersions

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Content

1

The motivation of the research topic

2

Introducing the biconservative immersions

3

Biharmonic and biconservative submanifolds

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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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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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. There are several ways to generalize these submanifolds:

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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. 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

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”.

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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. 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

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. Under the hypothesis of biconservativity some known results in the theory of submanifolds can be extended to more general contexts.

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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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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Biharmonic maps

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

• M |τ(φ)|2vg

Euler-Lagrange equation τ2(φ) = −∆φτ(φ)−traceg RN(dφ,τ(φ))dφ = 0. Critical points of E2 are called biharmonic maps.

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

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

• ∇φ∇φ −∇φ

∇

• is the rough Laplacian on 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, where ∆φ = −traceg

• ∇φ∇φ −∇φ

∇

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

RN(X,Y)Z = ∇N

X∇N Y Z −∇N Y ∇N XZ −∇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;

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The stress-bienergy tensor

G.Y. Jiang, 1987 defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2|τ(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 defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2|τ(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 defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2|τ(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 defined the stress-energy tensor S2 for the bienergy functional, and called it the stress-bienergy tensor: S2(X,Y) =1 2|τ(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 an isometric immersion then (divS2)♯ = −τ2(φ)⊤. In general, for an isometric immersion, divS2 = 0.

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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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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Biharmonic submanifolds; Biconservative submanifolds

Definition

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

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Biharmonic submanifolds; Biconservative submanifolds

Definition

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

Definition

A submanifold φ : Mm → Nn is called biconservative 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 ([Loubeau, Montaldo, Oniciuc – 2008])

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, where H = traceB/m ∈ C(NM) is the mean curvature vector field.

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Characterization results

Theorem ([Loubeau, Montaldo, Oniciuc – 2008])

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, where H = traceB/m ∈ C(NM) is the mean curvature vector field.

Proposition

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

1

M is biconservative;

2

traceA∇⊥

· H(·)+trace∇AH +trace

• RN(·,H)·

T = 0;

3

m 2 grad

• |H|2

+2traceA∇⊥

· H(·)+2trace

• RN(·,H)·

T = 0;

4

2trace∇AH − m

2 grad

• |H|2

= 0.

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Examples of biconservative submanifolds

Proposition

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

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Examples 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 ([Balmu¸ s, Montaldo, Oniciuc – 2013])

Let φ : Mm → Nn be a biconservative submanifold. Assume that M is pseudoumbilical, i.e., AH = |H|2I, and m = 4. Then M is CMC.

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Properties of biconservative submanifolds

Proposition ([Balmu¸ s, Montaldo, Oniciuc – 2013])

Let φ : Mm → Nn be a biconservative submanifold. Assume that M is pseudoumbilical, i.e., AH = |H|2I, and m = 4. Then M is CMC.

Proposition([N. – 2017])

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 pseudoumbilical.

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Properties of biconservative submanifolds

Proposition ([Balmu¸ s, Montaldo, Oniciuc – 2013])

Let φ : Mm → Nn be a biconservative submanifold. Assume that M is pseudoumbilical, i.e., AH = |H|2I, and m = 4. Then M is CMC.

Proposition([N. – 2017])

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 pseudoumbilical.

Proposition ([Montaldo, Oniciuc, Ratto – 2016])

Let φ : M2 → Nn be a biconservative surface. Then AH (∂z),∂z is holomorphic if and only if M is CMC.

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Characterization theorems

Theorem([Ou – 2010])

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 and f = traceA is the mean curvature function.

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Characterization theorems

Theorem([Ou – 2010])

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 and f = traceA is the mean curvature function. 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([Ou – 2010])

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 and f = traceA is the mean curvature function. 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

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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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Biconservative surfaces in N3(c)

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

Biconservative surfaces in N3(c)

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

Biconservative surfaces in N3(c)

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

Biconservative surfaces in N3(c)

Let φ : M2 → N3(c) be a non-CMC biconservative surface. Local results Global results extrinsic extrinsic intrinsic

Biconservative surfaces in N3(c)

Let φ : M2 → N3(c) be a non-CMC biconservative surface. Local results Global results extrinsic extrinsic intrinsic intrinsic

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Biconservative surfaces in N3(c)

Local conditions extrinsic intrinsic gradf = 0 on M c−K > 0 on M, gradK = 0 on M, and the level curves

• f K are certain circles

Biconservative surfaces in N3(c)

Local conditions extrinsic intrinsic gradf = 0 on M c−K > 0 on M, gradK = 0 on M, and the level curves

• f K are certain circles

Global conditions (M,g) complete and the above properties hold on an open and dense subset of M

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Local results

Theorem ([Caddeo, Montaldo, Oniciuc, Piu – 2014])

Let φ : M2 → N3(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 3K(c−K)2 = 0, where ∆ is the Laplace-Beltrami operator on M.

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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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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Local intrinsic characterization

Theorem ([Fetcu, N., Oniciuc – 2016])

Let

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

locally isometrically embedded in N3(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) .

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Local intrinsic characterization

Theorem ([Fetcu, N., Oniciuc – 2016])

Let

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

locally isometrically embedded in N3(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.

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Local intrinsic characterization

Theorem ([Fetcu, N., Oniciuc – 2016])

Let

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

locally isometrically embedded in N3(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. 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 N3(c) (up to an isometry of N3(c)), and not a

• ne-parameter family.

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Local intrinsic characterization

Theorem ([N., Oniciuc – 2017])

Let

• M2,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

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Local intrinsic characterization

Theorem ([N., Oniciuc – 2017])

Let

• M2,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 = e2ρ du2 +dv2 , ρ = ρ(u) satisfies ρ′′ = e−2ρ/3 −ce2ρ and ρ′ > 0;

u(ρ) =

ρ

ρ0

dτ

• −3e−2τ/3 −ce2τ +a

+u0,

where ρ is in some open interval I, ρ0 ∈ I and a,u0 ∈ R;

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Local intrinsic characterization

Theorem ([N., Oniciuc – 2017])

Let

• M2,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 = e2ρ du2 +dv2 , ρ = ρ(u) satisfies ρ′′ = e−2ρ/3 −ce2ρ and ρ′ > 0;

u(ρ) =

ρ

ρ0

dτ

• −3e−2τ/3 −ce2τ +a

+u0,

where ρ is in some open interval I, ρ0 ∈ I and a,u0 ∈ R; (ii) locally, g = e2ρ du2 +dv2 , ρ = ρ(u) satisfies 3ρ′′′ +2ρ′ρ′′ +8ce2ρρ′ = 0, ρ′ > 0 and c+e−2ρρ′′ > 0;

u(ρ) =

ρ

ρ0

dτ

• −3be−2τ/3 −ce2τ +a

+u0,

where ρ is in some open interval I, ρ0 ∈ I and a,b,u0 ∈ R, b > 0.

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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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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Local extrinsic results in R3

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Local extrinsic result in R3

Theorem ([Hasanis, Vlachos – 1995])

Let M2 be a surface in R3 with (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 in R3

Theorem ([Caddeo, Montaldo, Oniciuc, Piu – 2014])

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

C0(θ,v) =

• θ cosv,θ sinv,u ˜

C0(θ)

• ,

where u ˜

C0(θ) =

3 2 ˜ C0

• θ 1/3
• ˜

C0θ 2/3 −1+ 1 ˜ C0 log

• ˜

C0θ 1/3 +

• ˜

C0θ 2/3 −1

• with ˜

C0 a positive constant and θ ∈

• ˜

C−3/2 ,∞

• .

We denote X ˜

C0

• ˜

C−3/2 ,∞

• ×R
• = S ˜

C0.

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Global extrinsic results in R3

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Global extrinsic result in R3

Proposition ([Montaldo, Oniciuc, Ratto – 2016, N. – 2016])

If we consider the symmetry of Grafu ˜

C0, with respect to the Oθ(= Ox) axis, we

get a smooth, complete, biconservative surface ˜ S ˜

C0 in R3. Moreover, its mean

curvature function has its gradient grad ˜ f ˜

C0 is different from zero at any point of

an open dense subset of ˜ S ˜

C0.

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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 intrinsic results corresponding to c = 0

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Local intrinsic result; c = 0

Proposition ([N. – 2016])

Let

• M2,g = e2ρ

du2 +dv2 an abstract surface. Then g satisfies the local intrinisic conditions with c = 0 if and only if gC0 = C0(coshu)6 du2 +dv2 , where C0 > 0 is a constant.

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Global intrinsic results corresponding to c = 0

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Global intrinsic result; c = 0

Theorem ([N. – 2016])

Let

• R2,gC0 = C0 (coshu)6

du2 +dv2 . Then, we have:

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Global intrinsic result; c = 0

Theorem ([N. – 2016])

Let

• R2,gC0 = C0 (coshu)6

du2 +dv2 . Then, we have: (i)

• R2,gC0
• is complete;

(ii) the immersion φC0 :

• R2,gC0
• → R3 given by

φC0(u,v) =

• σ1

C0(u)cos(3v),σ1 C0(u)sin(3v),σ2 C0(u)

• is biconservative in R3, where

σ1

C0(u) =

√C0 3 (coshu)3 , σ2

C0(u) =

√C0 2 1 2 sinh(2u)+u

• ,

u ∈ R.

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Uniqueness

Theorem ([N., Oniciuc – 2017])

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

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Uniqueness

Theorem ([N., Oniciuc – 2017])

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

Theorem

Let M2 be a biconservative, complete and non-compact regular surface in R3. Then M = ˜ S ˜

C0.

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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 N3(c) Complete biconservative surfaces in R3 Complete biconservative surfaces in S3

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Local extrinsic results in S3

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Local extrinsic result in S3

Theorem ([Caddeo, Montaldo, Oniciuc, Piu – 2014])

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

Y ˜

C1(θ,v) =

• 1−
• 4

3√ ˜ C1 θ−3/4

2 cosµ(θ),

• 1−
• 4

3√ ˜ C1 θ−3/4

2 sinµ(θ),

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

• ,

(1)

where (θ,v) ∈ (θ01,θ02)×R, θ01 and θ02 are positive solutions of the equation −16 9 θ 2 −16θ 4 + ˜ C1θ 7/2 = 0 and µ(θ) = ±

θ

θ0 E(τ) dτ + ˜

ck, with ˜ ck ∈ R, k ∈ Z, and θ0 ∈ (θ01,θ02). If k = 0, we denote by S ˜

C1 = Y ˜ C1 ((θ01,θ02)×R).

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Global extrinsic results in S3

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Global extrinsic result in S3

The idea of the construction is to start with a surface S ˜

C1 and then to consider

Tk

• S ˜

C1

• , where Tk is a linear orthogonal transformation of R4 that acts on

span{e1,e2} as an axial orthogonal symmetry and leaves invariant span{e3,e4}, for k ∈ Z∗. We perform it infinitely many times.

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

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

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Using the stereographic projection, this construction can be illustrated in R3. N(0,0,0,1) k ∈ {−2,−1,0,1,2} N′(1,0,0,0) k ∈ {−2,−1,0,1,2}

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Local intrinsic results corresponding to c = 1

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Local intrinsic result; c = 1

Proposition ([N. – 2016])

Let

• M2,g
• be an abstract surface with g = e2ρ(u)(du2 +dv2), where u = u(ρ)

satisfies u =

ρ

ρ0

dτ

• −3be−2τ/3 −e2τ +a

+u0, where ρ 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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Theorem ([N. – 2016])

Let

• DC1,gC1
• . Then

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Theorem ([N. – 2016])

Let

• DC1,gC1
• . Then

(i)

• DC1,gC1
• is not complete;

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Theorem ([N. – 2016])

Let

• DC1,gC1
• . Then

(i)

• DC1,gC1
• is not complete;

(ii) 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 +ck, with ck ∈ R,

k ∈ Z, and ξ00 ∈ (ξ01,ξ02).

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Global intrinsic results corresponding to c = 1

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The key ingredient

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)

• ,

(2) where χ(ξ) = C∗

1/ξ,ν(ξ) = ±

ξ

ξ00 E(τ) dτ +c∗ k, C∗ 1 ∈

• 0,
• C1 −4/33/2−1/2

is a positive constant and c∗

k ∈ R, k ∈ Z.

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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,ck BICONSERVATIVE

S3

• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY φC1 = φ± C1,ck BICONSERVATIVE

S3

ψC1,C∗ 1 = ψ± C1,C∗ 1,c∗ k ISOMETRY

S±

C1,C∗

1,c∗ 1 ⊂ R3

• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY φC1 = φ± C1,ck BICONSERVATIVE

S3

ψC1,C∗ 1 = ψ± C1,C∗ 1,c∗ k 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 ∗ k a n d ±

• M2,g
• ξ01

ξ02 ξ θ

(DC1,gC1)

ISOMETRY φC1 = φ± C1,ck BICONSERVATIVE

S3

ψC1,C∗ 1 = ψ± C1,C∗ 1,c∗ k 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 ∗ k a n d ±

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

1 47 / 55

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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Open problem

Let φ : M2 → N3(c) be a biconservative surface. If φ is CMC on an open subset

• f M, then φ is CMC on M.

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Open problem

Let φ : M2 → N3(c) be a biconservative surface. If φ is CMC on an open subset

• f M, then φ is CMC on M.

φ : M2 → N3(c) is a biconservative surface if and only if A(gradf) = − f 2 gradf;

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Open problem

Let φ : M2 → N3(c) be a biconservative surface. If φ is CMC on an open subset

• f M, then φ is CMC on M.

φ : M2 → N3(c) is a biconservative surface if and only if A(gradf) = − f 2 gradf; if φ : M2 → N3(c) is a biconservative surface, then f∆f −3|gradf|2 −2A,Hessf = 0.

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Open problem

Let φ : M2 → N3(c) be a biconservative surface. If φ is CMC on an open subset

• f M, then φ is CMC on M.

φ : M2 → N3(c) is a biconservative surface if and only if A(gradf) = − f 2 gradf; if φ : M2 → N3(c) is a biconservative surface, then f∆f −3|gradf|2 −2A,Hessf = 0.

Open problem

Let M2 be a biconservative regular surface in S3. If M is compact, then is M a CMC surface?

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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 E4 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.

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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 R3 and S3, 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 R3 and S3, preprint, arXiv: 1701.07706.

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References III

[N., Oniciuc – 2017] S. Nistor, C. Oniciuc On the uniqueness of complete biconservative surfaces in R3 , work in progress. [Ou – 2010] Y.-L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math. 248 (2010), 217–232.

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Thank you!

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