On biconservative submanifoldsmath.etc.tuiasi.ro/.../Nistor-Presentation-Cagliari-2017.pdfApril 6,...

On biconservative submanifolds

Simona Nistor–Barna

“Alexandru Ioan Cuza” University of Iasi

Università degli Studi di Cagliari,April 6, 2017

1 / 88

Content

1 The motivation of the research topicGeneral contextHarmonic mapsBiharmonic maps

2 Properties of biconservative submanifoldsBiconservative submanifolds – Biharmonic submanifoldsBiconservative surfaces – Ricci surfacesLocal intrinsic characterization of biconservative surfaces in N3(c)Complete biconservative surfacesBiconservative surfaces in Nn

2 / 88

Content

2 / 88

Content

3 / 88

Content

4 / 88

General context

The study of submanifolds with constant mean curvature, i.e., CMCsubmanifolds, and of minimal submanifolds, represents a very active researchtopic in Differential Geometry for more than 50 years.

Examples of minimal surfaces

The plane The helicoid

Enneper’s surface The catenoid

5 / 88

General context

The study of submanifolds with constant mean curvature, i.e., CMCsubmanifolds, and of minimal submanifolds, represents a very active researchtopic in Differential Geometry for more than 50 years.

Examples of minimal surfaces

The plane The helicoid

Enneper’s surface The catenoid

5 / 88

General context

Examples of CMC surfaces

The sphere The cylinder

The nodoid The unduloid

6 / 88

General context

There are several ways to generalize these submanifolds:

the study of CMC submanifolds which satisfy some additional geometrichypotheses (CMC + biharmonicity);

the study of hypersurfaces in space forms, i.e., with constant sectionalcurvature, which are “highly non-CMC”.

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

7 / 88

General context

7 / 88

General context

7 / 88

General context

7 / 88

General context

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

The CMC hypersurfaces in space forms are trivially biconservative, so moreinteresting is the study of biconservative hypersurfaces which are non-CMC.

8 / 88

General context

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

The CMC hypersurfaces in space forms are trivially biconservative, so moreinteresting is the study of biconservative hypersurfaces which are non-CMC.

8 / 88

General context

The biconservative surfaces are closely related to the biharmonicsubmanifolds.

The biharmonic submanifolds represent a particular case of biharmonic maps,they being defined by Riemannian immersions which are also biharmonicmaps.

The biharmonic maps are critical points of the bienergy functional

E2 : C∞(M,N)→ R, E2(φ) =12

∫M‖τ(φ)‖2 vg,

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

9 / 88

General context

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

E2 : C∞(M,N)→ R, E2(φ) =12

∫M‖τ(φ)‖2 vg,

9 / 88

General context

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

E2 : C∞(M,N)→ R, E2(φ) =12

∫M‖τ(φ)‖2 vg,

9 / 88

Content

10 / 88

Harmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume that M iscompact and consider

The energy functional

E : C∞(M,N)→ R, E (φ) = E1 (φ) =12

∫M‖dφ‖2 vg.

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

∣∣∣∣t=0{E (φt)}= 0.

11 / 88

Harmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume that M iscompact and consider

The energy functional

E : C∞(M,N)→ R, E (φ) = E1 (φ) =12

∫M‖dφ‖2 vg.

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

∣∣∣∣t=0{E (φt)}= 0.

11 / 88

The first variation of the energy functional

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

The expression of the tension field in local charts

τ(φ) = gij

(∂ 2φ α

∂xi∂xj −M

∂φ α

∂xk + NΓ

∂φ β

∂xi∂φ δ

∂yα◦φ

where Γ represent the Christoffel symbols.

12 / 88

The first variation of the energy functional

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

The expression of the tension field in local charts

τ(φ) = gij

(∂ 2φ α

∂xi∂xj −M

∂φ α

∂xk + NΓ

∂φ β

∂xi∂φ δ

∂yα◦φ

where Γ represent the Christoffel symbols.

12 / 88

Harmonic maps

ExampleLet 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 ifand only if M is a minimal submanifold.

13 / 88

The stress-energy tensor

D. Hilbert, 1924, ([10]), associated to a functional E a symmetric tensorfield S of type (1,1), or (0,2), which is conservative at the critical points ofE, i.e., divS = 0 at these critical points, and called it the stress-energytensor.

To study harmonic maps, P. Baird si J. Eells, 1981; A. Sanini,1983,([1, 24]) used the tensor field

S =12‖dφ‖2g−φ

which satisfiesdivS =−〈τ(φ),dφ〉.

φ = harmonic⇒ divS = 0.

If φ is a submersion then divS = 0 if and only if φ is harmonic.

14 / 88

D. Hilbert, 1924, ([10]), associated to a functional E a symmetric tensorfield S of type (1,1), or (0,2), which is conservative at the critical points ofE, i.e., divS = 0 at these critical points, and called it the stress-energytensor.To study harmonic maps, P. Baird si J. Eells, 1981; A. Sanini,1983,([1, 24]) used the tensor field

S =12‖dφ‖2g−φ

14 / 88

S =12‖dφ‖2g−φ

14 / 88

S =12‖dφ‖2g−φ

14 / 88

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

It is not interesting to study Riemannian immersions with divS = 0.

15 / 88

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

It is not interesting to study Riemannian immersions with divS = 0.

15 / 88

The variational meaning of S

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

16 / 88

Content

17 / 88

Biharmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume M is compactand consider

The bienergy functional

E2 : C∞(M,N)→ R, E2 (φ) =12

∫M‖τ(φ)‖2 vg.

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

∣∣∣∣t=0{E2 (φt)}= 0.

18 / 88

Biharmonic maps

Let (Mm,g) and (Nn,h) be two Riemannian manifolds. Assume M is compactand consider

The bienergy functional

E2 : C∞(M,N)→ R, E2 (φ) =12

∫M‖τ(φ)‖2 vg.

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

∣∣∣∣t=0{E2 (φt)}= 0.

18 / 88

Biharmonic maps

Theorem ([11])A smooth map φ : (Mm,g)→ (Nn,h) is biharmonic if and only if the bitensionfield associated to φ ,

τ2(φ) =−∆φ

τ(φ)− traceg RN(dφ ,τ(φ))dφ ,

vanishes. Here,∆

φ =− traceg(∇

φ −∇φ

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

RN(X,Y)Z = ∇NX ∇

NY Z−∇

NY ∇

NX Z−∇

N[X,Y]Z.

19 / 88

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

τ2(φ) =−∆φ

τ(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;

20 / 88

τ2(φ) =−∆φ

20 / 88

τ2(φ) =−∆φ

20 / 88

τ2(φ) =−∆φ

20 / 88

The stress-bienergy tensor

G.Y. Jiang, 1987 ([12]), defined the stress-energy tensor S2 for thebienergy functional, and called it the stress-bienergy tensor:

S2(X,Y) =12‖τ(φ)‖2〈X,Y〉+ 〈dφ ,∇τ(φ)〉〈X,Y〉

−〈dφ(X),∇Yτ(φ)〉−〈dφ(Y),∇Xτ(φ)〉.

It satisfiesdivS2 =−〈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)

>. Ingeneral, for a Riemannian immersion, divS2 6= 0.

21 / 88

S2(X,Y) =12‖τ(φ)‖2〈X,Y〉+ 〈dφ ,∇τ(φ)〉〈X,Y〉

−〈dφ(X),∇Yτ(φ)〉−〈dφ(Y),∇Xτ(φ)〉.

21 / 88

S2(X,Y) =12‖τ(φ)‖2〈X,Y〉+ 〈dφ ,∇τ(φ)〉〈X,Y〉

−〈dφ(X),∇Yτ(φ)〉−〈dφ(Y),∇Xτ(φ)〉.

21 / 88

S2(X,Y) =12‖τ(φ)‖2〈X,Y〉+ 〈dφ ,∇τ(φ)〉〈X,Y〉

−〈dφ(X),∇Yτ(φ)〉−〈dφ(Y),∇Xτ(φ)〉.

21 / 88

The variational meaning of S2

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

22 / 88

Content

23 / 88

Basic facts in the submanifolds theory

Let φ : Mm→ Nn be a submanifold.

Locally,dφ(X)≡ X; ∇

Xdφ(Y)≡ ∇NX Y;

Globally,

φ−1(TN) =

⋃p∈M

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

TM ≡⋃

dφp (TpM) ; NM =⋃

dφp (TpM)⊥ .

24 / 88

Xdφ(Y)≡ ∇NX Y;

Globally,

φ−1(TN) =

⋃p∈M

TM ≡⋃

dφp (TpM)⊥ .

24 / 88

Xdφ(Y)≡ ∇NX Y;

Globally,

φ−1(TN) =

⋃p∈M

TM ≡⋃

dφp (TpM)⊥ .

24 / 88

The Gauss equation∇

NX Y = ∇XY +B(X,Y);

The Weingarten equation

∇NX η =−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 = fm η , f is the (m-times)

mean curvature function.

25 / 88

⊥X η ;

H =traceB

m∈ C(NM);

25 / 88

⊥X η ;

〈B(X,Y),η〉= 〈Aη(X),Y〉;

H is the mean curvature vector field

H =traceB

m∈ C(NM);

25 / 88

⊥X η ;

H =traceB

m∈ C(NM);

25 / 88

⊥X η ;

H =traceB

m∈ C(NM);

25 / 88

Content

26 / 88

Biconservative submanifolds; Biharmonicsubmanifolds

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

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

> = 0.

27 / 88

Biconservative submanifolds; Biharmonicsubmanifolds

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

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

> = 0.

27 / 88

Mm submanifold of Nn

Mm biconservative

Mm biharmonic

Mm minimal

28 / 88

Mm biconservative

Mm biharmonic

Mm minimal

28 / 88

Mm biconservative

Mm biharmonic

Mm minimal

28 / 88

Mm biconservative

Mm biharmonic

Mm minimal

28 / 88

Characterization results

Theorem ([5, 13, 21])A submanifold φ : Mm→ Nn is biharmonic if and only if

traceA∇⊥· H(·)+ trace∇AH + trace

(RN(·,H)·

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 m2 grad

(‖H‖2

)+2traceA

∇⊥· H(·)+2trace(RN(·,H)·

)T= 0;

4 2trace∇AH− m2 grad

(‖H‖2

29 / 88

Characterization results

Theorem ([5, 13, 21])A submanifold φ : Mm→ Nn is biharmonic if and only if

traceA∇⊥· H(·)+ trace∇AH + trace

(RN(·,H)·

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 m2 grad

(‖H‖2

)+2traceA

∇⊥· H(·)+2trace(RN(·,H)·

)T= 0;

4 2trace∇AH− m2 grad

(‖H‖2

29 / 88

Properties of biconservative submanifolds

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

PropositionLet φ : Mm→ Nn be a submanifold. If N is a space form, i.e., has constantsectional curvature, and M is PMC, i.e., has H parallel in NM, then M isbiconservative.

Proposition ([2])Let φ : Mm→ Nn be a submanifold. Assume that M is pseudoumbilical, i.e.,AH = ‖H‖2I, and m 6= 4. Then M is CMC.

30 / 88

Characterization theorems

Theorem([2, 22])If φ : Mm→ Nm+1 is a hypersurface, then M is biharmonic if and only if

2A(grad f )+ f grad f −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(grad f ) =− f2

grad f .

Every CMC hypersurface in Nm+1(c) is biconservative.

31 / 88

)T= 0,

A(grad f ) =− f2

grad f .

31 / 88

)T= 0,

A(grad f ) =− f2

grad f .

31 / 88

Content

32 / 88

Biconservative surfaces

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

Local proprieties Global properties

f > 0 and grad f 6= 0on M

f > 0 on M and grad f 6= 0on an open and dense

subset of M

extrinsic intrinsic extrinsic intrinsic

33 / 88

subset of M

33 / 88

subset of M

33 / 88

subset of M

extrinsic intrinsic

33 / 88

subset of M

33 / 88

Local extrinsic properties

Theorem ([3])Let φ : M2→ N3(c) be a biconservative surface with f > 0 and grad f 6= 0 at anypoint in M. Then we have

f ∆f + |grad f |2 + 43

cf 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 alocal chart (U;u,v) such that f = f (u,v) = f (u) and (1) is equivalent with

ff ′′− 74(f ′)2− 4

3cf 2 + f 4 = 0, (2)

i.e., f has to satisfy a second order ODE.

34 / 88

Local extrinsic properties

Theorem ([3])Let φ : M2→ N3(c) be a biconservative surface with f > 0 and grad f 6= 0 at anypoint in M. Then we have

f ∆f + |grad f |2 + 43

cf 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 alocal chart (U;u,v) such that f = f (u,v) = f (u) and (1) is equivalent with

ff ′′− 74(f ′)2− 4

3cf 2 + f 4 = 0, (2)

i.e., f has to satisfy a second order ODE.

34 / 88

Local intrinsic properties

Using the Gauss equation, K = c+detA, we get

f 2 =43(c−K). (3)

TheoremLet φ : M2→ N3(c) be a biconservative surface with f > 0 and grad f 6= 0 at anypoint of M. Then we obtain

(c−K)∆K−|gradK|2− 83

K(c−K)2 = 0, (4)

where ∆ is Laplace-Beltrami operator on M.

35 / 88

Using the Gauss equation, K = c+detA, we get

f 2 =43(c−K). (3)

TheoremLet φ : M2→ N3(c) be a biconservative surface with f > 0 and grad f 6= 0 at anypoint of M. Then we obtain

(c−K)∆K−|gradK|2− 83

K(c−K)2 = 0, (4)

where ∆ is Laplace-Beltrami operator on M.

35 / 88

The intrinsic problem

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

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

with f > 0 and grad f 6= 0.

36 / 88

Ricci problem

Given an abstract surface(M2,g

), we want to determine the necessary

and sufficient conditions such that it admits, locally, a minimal embeddingin 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 aminimal 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 theRicci condition. If (5) holds, then, locally, M admits a one-parametricfamily of minimal embeddings in N3(c).

37 / 88

Ricci problem

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

(c−K)∆K−|gradK|2−4K(c−K)2 = 0. (5)

37 / 88

Ricci problem

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

(c−K)∆K−|gradK|2−4K(c−K)2 = 0. (5)

37 / 88

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

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

)bicons. in R3

)satisfies (4), K < 0

(M2,g1/2 =

√−Kg

)Ricci, K1/2 < 0

)Ricci, K < 0

(M2,g−1 = (−K)−1 g

)satisfies (4), K−1 < 0

38 / 88

)bicons. in R3

(M2,g1/2 =

√−Kg

)Ricci, K1/2 < 0

)Ricci, K < 0

(M2,g−1 = (−K)−1 g

38 / 88

)bicons. in R3

(M2,g1/2 =

√−Kg

)Ricci, K1/2 < 0

)Ricci, K < 0

(M2,g−1 = (−K)−1 g

38 / 88

)bicons. in R3

(M2,g1/2 =

√−Kg

)Ricci, K1/2 < 0

)Ricci, K < 0

(M2,g−1 = (−K)−1 g

38 / 88

The link between the biconservativity and the Riccicondition with respect to c

Theorem ([7])Let

)be a biconservative surface in N3(c). If f > 0 and grad f 6= 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 +∆

log(c−Kr)+r2

log(c−K)

and Kr, which denotes the Gaussian curvature on M corresponding to themetric (c−K)rg, is given by

Kr = (c−K)−r(

3−4r3

log(c−K)∆r+(c−K)−1g(gradr,gradK)

39 / 88

Content

40 / 88

Local intrinsic characterization of biconservativesurfaces in N3(c)

Theorem ([7])Let

)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 andgrad f 6= 0 at any point if and only if c−K > 0, gradK 6= 0, at any point, and itslevel curves are circles in M with constant curvature

κ =3|gradK|8(c−K)

CorollaryIf the surface M is simply connected, then the theorem holds globally, but, inthis case, instead of a local isometric embedding we have a global isometricimmersion.

41 / 88

Local intrinsic characterization of biconservativesurfaces in N3(c)

Theorem ([7])Let

)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 andgrad f 6= 0 at any point if and only if c−K > 0, gradK 6= 0, at any point, and itslevel curves are circles in M with constant curvature

CorollaryIf the surface M is simply connected, then the theorem holds globally, but, inthis case, instead of a local isometric embedding we have a global isometricimmersion.

41 / 88

Local intrinsic characterization

Theorem ([19])Let

)be an abstract surface with c−K(p)> 0 and (gradK)(p) 6= 0 at any point

p ∈M, where c ∈ R is a constant. Let X1 = (gradK)/|gradK| and X2 ∈ C(TM) be twovector fields on M such that {X1(p),X2(p)} is a positively oriented basis at any point ofp ∈M. Then, the following conditions are equivalent:

(a) the level curves of K are circles in M with constant curvature

8(c−K);

X2 (X1K) = 0 and ∇X2 X2 =−3X1K8(c−K)

(c) locally, the metric g can be written as g = (c−K)−3/4 (du2 +dv2), where (u,v) arelocal coordinates positively oriented, K = K(u), and K′ > 0;

42 / 88

Theorem ([19])Let

8(c−K);

X2 (X1K) = 0 and ∇X2 X2 =−3X1K8(c−K)

42 / 88

Theorem ([19])Let

8(c−K);

X2 (X1K) = 0 and ∇X2 X2 =−3X1K8(c−K)

42 / 88

(d) locally, the metric g can be written as g = e2ϕ(du2 +dv2

), where (u,v) are local coordinates positively

oriented, 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 =∫

dτ√−3e−2τ/3− ce2τ +a

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

3ϕ′′′+2ϕ

′ϕ′′+8ce2ϕ

ϕ′ = 0 (7)

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

u =∫

dτ√−3be−2τ/3− ce2τ +a

where ϕ is in some open interval I and a,b,u0 ∈ R are constants, b > 0.

43 / 88

(d) locally, the metric g can be written as g = e2ϕ(du2 +dv2

ϕ′′ = e−2ϕ/3− ce2ϕ (6)

and the condition ϕ ′ > 0; moreover, the solutions of the above equation, u = u(ϕ), are

u =∫

dτ√−3e−2τ/3− ce2τ +a

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

3ϕ′′′+2ϕ

′ϕ′′+8ce2ϕ

ϕ′ = 0 (7)

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

u =∫

dτ√−3be−2τ/3− ce2τ +a

where ϕ is in some open interval I and a,b,u0 ∈ R are constants, b > 0.

43 / 88

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 ofthe parameter with α > 0 (u = α u+β ), are given by

ϕ(u) = 3log(coshu)+ constant, u > 0.

44 / 88

RemarksIf 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 6= 0, then we cannot determine explicitly ϕ = ϕ(u), but we haveu = u(ϕ). Another way to see that we have only a one-parameter family ofsolutions of equation (7) is to rewrite the metric g in certainnon-isothermal coordinates.

Further, we consider only the c = 1 case.

45 / 88

(du2 +dv2

), C0 > 0.

45 / 88

(du2 +dv2

), C0 > 0.

45 / 88

Proposition ([17])Let

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

satisfiesu =

dτ√−3be−2τ/3− e2τ +a

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

)is isometric to(

DC1 ,gC1 =3

ξ 2(−ξ 8/3 +3C1ξ 2−3

)dξ2 +

1ξ 2 dθ

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, with0 < ξ01 < ξ02.

46 / 88

RemarkLet us consider(

DC1 ,gC1 =3

ξ 2(−ξ 8/3 +3C1ξ 2−3

)dξ2 +

1ξ 2 dθ

and DC′1,gC′1

ξ 2(−ξ 8/3 +3C′1ξ 2−3

)dξ2 +

ξ 2dθ

The surfaces(DC1 ,gC1

(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 aone-parameter family of surfaces.

47 / 88

Content

48 / 88

Complete biconservative surfaces in R3

49 / 88

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 grad f 6= 0 at any point of anopen and dense subset.from intrinsic point of view, we construct a complete abstract surface(M2,g

)with K < 0 on M and gradK 6= 0 on an open and dense subset of

M, which admits a biconservative immersion in R3, defined on whole thesurface M, with f > 0 on M and grad f 6= 0 on that open and dense subsetof M.

50 / 88

with f > 0 at any point of the surface and grad f 6= 0 at any point of anopen and dense subset.

from intrinsic point of view, we construct a complete abstract surface(M2,g

50 / 88

with f > 0 at any point of the surface and grad f 6= 0 at any point of anopen and dense subset.from intrinsic point of view, we construct a complete abstract surface(M2,g

50 / 88

51 / 88

Local extrinsic result

Theorem ([9])Let M2 be a surface in R3 with f (p)> 0 and (grad f )(p) 6= 0 at any p ∈M. Then,M2 is biconservative if and only if, locally, it is a surface of revolution, and thecurvature κ = κ(u) of the profile curve σ = σ(u), ‖σ ′(u)‖= 1, is positivesolution of the following ODE

κ′′κ =

74(κ′)2−4κ

52 / 88

Theorem ([3])Let M2 be a biconservative surface in R3 with f (p)> 0 and (grad f )(p) 6= 0 atany p ∈M. Then, locally, the surface can be parametrized by

XC0(ρ,v) =

(ρ cosv,ρ sinv,uC0

(ρ)),

uC0(ρ) =

1/3√

C0ρ2/3−1+1√C0

log(√

C0ρ1/3 +

√C0ρ2/3−1

with C0 a positive constant and ρ ∈(

C−3/20 ,∞

53 / 88

Any two such surfaces are not locally isometric, so we have aone-parameter family of biconservative surfaces in R3.

These surfaces are NOT complete.

We denote by SC0the image XC0

((C−3/2

0 ,∞)×R

). The boundary of SC0

i.e., SC0\SC0

, is the circle(

C−3/20 cosv, C−3/2

0 sinv,0)

, which lies in the xOy

plane. At a boundary point, the tangent plane to the closure SC0of SC0

isparallel to Oz. Moreover, along the boundary, the mean curvature functionis constant fC0

2C3/20

)/3 and grad fC0

Thus, in order to obtain a complete biconservative surface in R3, weproved that we can glue two biconservative surfaces SC0

and SC′0, at the

level of C∞ smoothness, only along the boundary and, in this case,C0 = C′0.

54 / 88

Any two such surfaces are not locally isometric, so we have aone-parameter family of biconservative surfaces in R3.These surfaces are NOT complete.

((C−3/2

0 ,∞)×R

i.e., SC0\SC0

, is the circle(

C−3/20 cosv, C−3/2

0 sinv,0)

2C3/20

)/3 and grad fC0

and SC′0, at the

54 / 88

((C−3/2

0 ,∞)×R

i.e., SC0\SC0

, is the circle(

C−3/20 cosv, C−3/2

0 sinv,0)

2C3/20

)/3 and grad fC0

and SC′0, at the

54 / 88

((C−3/2

0 ,∞)×R

i.e., SC0\SC0

, is the circle(

C−3/20 cosv, C−3/2

0 sinv,0)

2C3/20

)/3 and grad fC0

and SC′0, at the

54 / 88

Global extrinsic result

Proposition ([15, 17])If we consider the symmetry of GrafuC, with respect to the Oρ(= Ox) axis, weget a smooth, complete, biconservative surface SC0

in R3. Moreover, its meancurvature function fC0

is positive and grad fC0is different from zero at any point

of an open dense subset of SC0.

55 / 88

Uniqueness

Theorem ([20])Let M2 be a biconservative regular surface in R3. If M is compact, then M isCMC.

TheoremLet M2 be a biconservative regular surface in R3. Assume that M is acomplete, non-compact surface and the number of the connectedcomponents of

W = {p ∈M | (grad f )(p) 6= 0}

is finite. Then M = SC0.

56 / 88

Uniqueness

Theorem ([20])Let M2 be a biconservative regular surface in R3. If M is compact, then M isCMC.

TheoremLet M2 be a biconservative regular surface in R3. Assume that M is acomplete, non-compact surface and the number of the connectedcomponents of

W = {p ∈M | (grad f )(p) 6= 0}

is finite. Then M = SC0.

56 / 88

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

57 / 88

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 6= 0 at any point of R2 \Ov;

(c) the immersion ϕC0 :(R2,gC0

)→ R3 given by

ϕC0 (u,v) =(

σ1C0(u)cos(3v),σ 1

C0(u)sin(3v),σ 2

C0(u))

is biconservative in R3, where

σ1C0(u) =

3(coshu)3 , σ

2C0(u) =

sinh(2u)+u), u ∈ R.

58 / 88

Theorem ([17])Let

KC0 (u,v) = KC0 (u) =−3

C0 (coshu)8 < 0, K′C0(u) =

24sinhu

C0 (coshu)9 ,

)→ R3 given by

ϕC0 (u,v) =(

C0(u)sin(3v),σ 2

C0(u))

σ1C0(u) =

3(coshu)3 , σ

2C0(u) =

58 / 88

Theorem ([17])Let

KC0 (u,v) = KC0 (u) =−3

C0 (coshu)8 < 0, K′C0(u) =

24sinhu

C0 (coshu)9 ,

)→ R3 given by

ϕC0 (u,v) =(

C0(u)sin(3v),σ 2

C0(u))

σ1C0(u) =

3(coshu)3 , σ

2C0(u) =

58 / 88

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.

PropositionLet

(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

of revolution in R3 given by

Z(u,v) =

(α(u)cosh

),α(u)sinh

),β (u)

), (u,v) ∈ R2,

where a ∈ (0,√

3] and

α(u) =a

coshu, β (u) =

√(3−a2)cosh2

τ +a2

cosh2τ

59 / 88

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.

PropositionLet

(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

of revolution in R3 given by

Z(u,v) =

(α(u)cosh

),α(u)sinh

),β (u)

), (u,v) ∈ R2,

where a ∈ (0,√

3] and

α(u) =a

coshu, β (u) =

√(3−a2)cosh2

τ +a2

cosh2τ

59 / 88

RemarkWhen a =

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

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 followingnon-existence result.

Theorem ([4, 6])There exists no proper biharmonic surface in R3.

60 / 88

RemarkWhen a =

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

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 followingnon-existence result.

Theorem ([4, 6])There exists no proper biharmonic surface in R3.

60 / 88

Complete biconservative surfaces in S3

61 / 88

In this section we consider the global problem and construct biconservativesurfaces in S3 with f > 0 at any point of the surface and grad f 6= 0 at any pointof an open and dense subset.

62 / 88

)ξ01 ξ02 ξ

(DC1 ,gC1)

ISOMETRY

φ ±C1,c1

RVATIV

ψC1 ,C∗1= ψ±C1 ,C

∗1 ,c∗1

ISOMETRY

S±C1,C∗1 ,c∗1⊂ R3

SC1,C∗1 ⊂ R3 complete

playing with the const.c ∗1 and ±

playing withthe const. c1 and ±,

ΦC1,C∗1

63 / 88

)ξ01 ξ02 ξ

(DC1 ,gC1)

ISOMETRY

φ ±C1,c1

RVATIV

ψC1 ,C∗1= ψ±C1 ,C

∗1 ,c∗1

ISOMETRY

S±C1,C∗1 ,c∗1⊂ R3

ΦC1,C∗1

63 / 88

)ξ01 ξ02 ξ

(DC1 ,gC1)

ISOMETRY

φ ±C1,c1

RVATIV

ψC1 ,C∗1= ψ±C1 ,C

∗1 ,c∗1

ISOMETRY

S±C1,C∗1 ,c∗1⊂ R3

ΦC1,C∗1

63 / 88

)ξ01 ξ02 ξ

(DC1 ,gC1)

ISOMETRY

φ ±C1,c1

RVATIV

ψC1 ,C∗1= ψ±C1 ,C

∗1 ,c∗1

ISOMETRY

S±C1,C∗1 ,c∗1⊂ R3

ΦC1,C∗1

63 / 88

)ξ01 ξ02 ξ

(DC1 ,gC1)

ISOMETRY

φ ±C1,c1

RVATIV

ψC1 ,C∗1= ψ±C1 ,C

∗1 ,c∗1

ISOMETRY

S±C1,C∗1 ,c∗1⊂ R3

ΦC1,C∗1

63 / 88

The projection of ΦC1,C∗1on the Ox1x2 plane is a curve which lies in the

annulus of radii√

1−1/(C1ξ 2

√1−1/

(C1ξ 2

). It has self-intersections

and is dense in the annulus. Choosing C1 = C∗1 = 1, we obtain

64 / 88

The signed curvature of the profile curve of SC1,C∗1.

65 / 88

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

66 / 88

Theorem ([3])Let M2 be a biconservative surface in S3 with f (p)> 0 and (grad f )(p) 6= 0 for any p ∈M. Then, locally, the surfaceviewed in R4, can be parametrized by

YC1(u,v) = σ(u)+

4κ(u)−3/4

(f 1(cosv−1)+ f 2 sinv

where C1 ∈(64/

),∞)

is a positive constant; f 1, f 2 ∈ R4 are two constant orthonormal vectors; σ(u) is acurve parametrized by arclength that satisfies

〈σ(u), f 1〉=4κ(u)−3/4

, 〈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 =

74(κ ′)

κ2−4κ

such that(κ ′)

2=− 16

2−16κ4 + C1κ

67 / 88

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

YC1(κ,v) =

(√1−(

C1κ−3/4

)2cos µ(κ),

√1−(

C1κ−3/4

)2sin µ(κ),

C1κ−3/4 cosv, 4

C1κ−3/4 sinv

where (κ,v) ∈ (κ01,κ02)×R, κ01 and κ02 are positive solutions of the equation

−169

κ2−16κ

4 + C1κ7/2 = 0

and µ(κ) =±∫

κ0E(τ) dτ + c0, with c0 ∈ R and κ0 ∈ (κ01,κ02).

We choose κ0 = (3C1/64)2.

An alternative expression for YC1was given in [8].

68 / 88

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

The surface(DC1 ,gC1

)defined in the previous subsection is NOT complete but

it has the following properties.

69 / 88

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

The surface(DC1 ,gC1

)defined in the previous subsection is NOT complete but

it has the following properties.

69 / 88

Theorem ([17])Let (DC1 ,gC1). Then

(a) KC1(ξ ,θ) = K(ξ ,θ),

1−K(ξ ,θ) =19

ξ8/3 > 0, K′(ξ ) =− 8

and gradK 6= 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

ξ00E(τ) dτ + c1, with c1 ∈ R si

ξ00 ∈ (ξ01,ξ02).

Alegem ξ00 = (9C1/4)3/2.

70 / 88

(a) KC1(ξ ,θ) = K(ξ ,θ),

1−K(ξ ,θ) =19

ξ8/3 > 0, K′(ξ ) =− 8

φC1 (ξ ,θ) =

(√1− 1

C1ξ 2 cosζ (ξ ),

√1− 1

C1θ)√C1ξ

,sin(√

C1θ)√C1ξ

ξ00 ∈ (ξ01,ξ02).

Alegem ξ00 = (9C1/4)3/2.

70 / 88

(a) KC1(ξ ,θ) = K(ξ ,θ),

1−K(ξ ,θ) =19

ξ8/3 > 0, K′(ξ ) =− 8

φC1 (ξ ,θ) =

(√1− 1

C1ξ 2 cosζ (ξ ),

√1− 1

C1θ)√C1ξ

,sin(√

C1θ)√C1ξ

ξ00 ∈ (ξ01,ξ02).

Alegem ξ00 = (9C1/4)3/2.

70 / 88

(a) KC1(ξ ,θ) = K(ξ ,θ),

1−K(ξ ,θ) =19

ξ8/3 > 0, K′(ξ ) =− 8

φC1 (ξ ,θ) =

(√1− 1

C1ξ 2 cosζ (ξ ),

√1− 1

C1θ)√C1ξ

,sin(√

C1θ)√C1ξ

ξ00 ∈ (ξ01,ξ02).

Alegem ξ00 = (9C1/4)3/2.

70 / 88

The key ingredient

The key ingredient in the construction of the complete biconservative surfacein 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 ofrevolution in R3, which on an open and dense subset is locally isometric with(DC1 ,gC1

71 / 88

TheoremLet

(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

C∗1,χ(ξ )sin

C∗1,ν(ξ )

), (9)

where χ(ξ ) = C∗1/ξ ,ν(ξ ) =±∫ ξ

ξ00E(τ) dτ + c∗1, C∗1 ∈

(0,(C1−4/33/2

)−1/2)

is apositive constant and c∗1 ∈ R.

72 / 88

Content

73 / 88

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) thatsatisfies divT = 0 and then we focus on the biconservative case.

74 / 88

Properties of T

Theorem ([18])Let

)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.

75 / 88

Properties of T

Theorem ([18])Let

any two relations involve any of the others1 divT = 0;

2 t is constant, where t = traceT;3 〈T (∂z) ,∂z〉 is holomorphic;4 T is a Codazzi tensor field.

75 / 88

Properties of T

Theorem ([18])Let

any two relations involve any of the others1 divT = 0;2 t is constant, where t = traceT;

3 〈T (∂z) ,∂z〉 is holomorphic;4 T is a Codazzi tensor field.

75 / 88

Properties of T

Theorem ([18])Let

any two relations involve any of the others1 divT = 0;2 t is constant, where t = traceT;3 〈T (∂z) ,∂z〉 is holomorphic;

4 T is a Codazzi tensor field.

75 / 88

Properties of T

Theorem ([18])Let

any two relations involve any of the others1 divT = 0;2 t is constant, where t = traceT;3 〈T (∂z) ,∂z〉 is holomorphic;4 T is a Codazzi tensor field.

75 / 88

T = S2

M2 bicons.

|H| const.

〈S2 (∂z) ,∂z〉holomorphic

S2 Codazzi

T = AH

divAH = 0

|H| const.

〈AH (∂z) ,∂z〉holomorphic

AH Codazzi

76 / 88

T = S2

M2 bicons.

|H| const.

〈S2 (∂z) ,∂z〉holomorphic

S2 Codazzi T = AH

divAH = 0

|H| const.

〈AH (∂z) ,∂z〉holomorphic

AH Codazzi

76 / 88

We recall that for a surface

S2 =−2|H|2I +4AH,

therefore, in general,

S2 is Codazzi 6⇔ AH is CodazzidivS2 = 0 6⇔ divAH = 0.

divS2 =−2grad(|H|2

)+4divAH.

77 / 88

Properties of biconservative surfaces

TheoremLet 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.

CorollaryLet 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.

78 / 88

1 M is a biconservative surface;

2 |H| is constant;3 〈AH (∂z) ,∂z〉 is holomorphic;4 AH is a Codazzi tensor field.

78 / 88

1 M is a biconservative surface;2 |H| is constant;

3 〈AH (∂z) ,∂z〉 is holomorphic;4 AH is a Codazzi tensor field.

78 / 88

1 M is a biconservative surface;2 |H| is constant;3 〈AH (∂z) ,∂z〉 is holomorphic;

4 AH is a Codazzi tensor field.

78 / 88

1 M is a biconservative surface;

2 AH is a Codazzi tensor field;3 〈AH (∂z) ,∂z〉 is holomorphic.

78 / 88

1 M is a biconservative surface;2 AH is a Codazzi tensor field;

3 〈AH (∂z) ,∂z〉 is holomorphic.

78 / 88

Rough Laplacian ∆RS2

PropositionLet

)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 +∇grad t, (10)

where ∆RT =− trace(∇2T

), t = traceT and I is the identity tensor field of type

(1,1).

CorollaryIf 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.

79 / 88

PropositionLet

∆RT =−2KT + tKI +(∆t)I +∇grad t, (10)

(1,1).

(|τ(ϕ)|2

K|τ(ϕ)|2 +∆|τ(ϕ)|2)

I, (11)

79 / 88

PropositionLet

∆RT =−2KT + tKI +(∆t)I +∇grad t, (10)

(1,1).

(|τ(ϕ)|2

K|τ(ϕ)|2 +∆|τ(ϕ)|2)

I, (11)

79 / 88

The Simons type formula for S2

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

12 ∆ |S2|2 = −2K |S2|2 +div

((〈S2,grad

(|τ(ϕ)|2

)〉)])

+K|τ(ϕ)|4

+ 12 ∆(|τ(ϕ)|4

)+∣∣grad

(|τ(ϕ)|2

)∣∣2−|∇S2|2. (12)

80 / 88

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

(|∇S2|2 +2K

(|S2|2−

|τ(ϕ)|4

))vg =

∣∣∣grad(|τ(ϕ)|2

)∣∣∣2 vg,

or, equivalent,∫M

(|∇AH |2 +2K

(|AH |2−2|H|4

))vg =

∣∣∣grad(|H|2

)∣∣∣2 vg.

81 / 88

Properties of biconservative surfaces in Nn

M2 surface, ∇AH = 0 M2 bicons.

M2 is CMC, compact, K ≥ 0

TheoremLet ϕ : M2→ Nn be a CMC biconservative surface and assume that M iscompact. If K ≥ 0, then ∇AH = 0 and M is flat or M is pseudoumbilical.

82 / 88

Properties of biconservative surfaces in Nn

M2 surface, ∇AH = 0 M2 bicons.

M2 is CMC, compact, K ≥ 0

TheoremLet ϕ : M2→ Nn be a CMC biconservative surface and assume that M iscompact. If K ≥ 0, then ∇AH = 0 and M is flat or M is pseudoumbilical.

82 / 88

References I

[1] P. Baird, J. Eells,A conservation law for harmonic maps, Geometry Symposium Utrecht1980, 1–25, Lecture Notes in Math. 894, Springer, Berlin-New York,1981.

[2] A. Balmus, 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-freestress-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 PureMathematics, 1. World Scientific Publishing Co., Singapore, 1984.

83 / 88

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-spaceforms, 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.

84 / 88

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 Riemannianmanifolds, 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.

85 / 88

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. OniciucGlobal properties of biconservative surfaces in R3 and S3, accepted.

[20] S. Nistor-Barna, C. OniciucOn the uniqueness of complete biconservative surfaces in R3 , work inprogress.

86 / 88

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 adeformazioni di metriche, Rend. Mat. 3 (1983), 53–63.

87 / 88

Thank you!

88 / 88

On biconservative submanifoldsmath.etc.tuiasi.ro/.../Nistor-Presentation-Cagliari-2017.pdfApril 6,...

Documents

Transcript of On biconservative submanifoldsmath.etc.tuiasi.ro/.../Nistor-Presentation-Cagliari-2017.pdfApril 6,...