On biconservative surfacesmath.etc.tuiasi.ro/dfetcu/resurse/Fetcu-Brno_July_2016_short.pdf ·...

On biconservative surfaces

Dorel Fetcu

Gheorghe Asachi Technical University of Iasi, Romania

Brno, Czechia, July 2016

Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 1 / 30

The harmonic and biharmonic problems

Harmonic and biharmonic maps

Let ϕ : (M,g)→ (N,h) be a smooth map.

Energy functional

E (ϕ) = E1 (ϕ) =12

∫M|dϕ|2vg

Euler-Lagrange equation

τ(ϕ) = τ1(ϕ) = traceg∇dϕ

= 0

Critical points of E:harmonic maps

Bienergy functional

E2 (ϕ) =12

∫M|τ(ϕ)|2vg

Euler-Lagrange equation

τ2(ϕ) = −∆ϕ

τ(ϕ)− traceg RN(dϕ,τ(ϕ))dϕ

= 0

Critical points of E2:biharmonic maps


The harmonic and biharmonic problems The biharmonic equation

The biharmonic equation (Jiang, 1986)

τ2(ϕ) =−∆ϕ

τ(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 = ∇NX ∇

NY Z−∇

NY ∇

NX Z−∇

N[X,Y]Z

is a fourth-order non-linear elliptic equationany harmonic map is biharmonica non-harmonic biharmonic map is called proper biharmonica submanifold M of a Riemannian manifold N is called abiharmonic submanifold if the immersion ϕ : M→ N is biharmonic(ϕ is harmonic if and only if M is minimal)


Introducing the biconservative submanifolds History

Biconservative submanifolds

D. Hilbert, 1924, described a symmetric 2-covariant tensor S,associated to a variational problem, conservative at critical points,i.e., S satisfies divS = 0 at these points, and called it thestress-energy tensor

P. Baird and J. Eells, 1981; A. Sanini, 1983, used the tensor

S =12|dϕ|2g−ϕ

∗h

that satisfiesdivS =−〈τ(ϕ),dϕ〉,

to study harmonic maps, since

ϕ = harmonic⇒ divS = 0

Obviously

ϕ : M→ N is an isometric immersion⇒ τ(ϕ) = normal⇒ divS = 0



Biconservative submanifolds

D. Hilbert, 1924, described a symmetric 2-covariant tensor S,associated to a variational problem, conservative at critical points,i.e., S satisfies divS = 0 at these points, and called it thestress-energy tensorP. Baird and J. Eells, 1981; A. Sanini, 1983, used the tensor

S =12|dϕ|2g−ϕ

∗h

that satisfiesdivS =−〈τ(ϕ),dϕ〉,

to study harmonic maps, since

ϕ = harmonic⇒ divS = 0

Obviously

ϕ : M→ N is an isometric immersion⇒ τ(ϕ) = normal⇒ divS = 0



G. Y. Jiang, 1987, defined the stress-energy tensor S2 of thebienergy:

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

−〈dϕ(X),∇Yτ(ϕ)〉−〈dϕ(Y),∇Xτ(ϕ)〉

that satisfiesdivS2 =−〈τ2(ϕ),dϕ〉

If ϕ : M→ N is an isometric immersion, then divS2 =−τ2(ϕ)>

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


Introducing the biconservative submanifolds Definition

DefinitionA submanifold ϕ : M→ N of a Riemannian manifold N is called abiconservative submanifold if divS2 = 0, i.e., τ2(ϕ)

> = 0.

Theorem (Balmus-Montaldo-Oniciuc, 2012)

A submanifold M in a Riemannian manifold N, with secondfundamental form σ , mean curvature vector field H, and shapeoperator A, is biharmonic if and only if

m2

∇|H|2 +2traceA∇⊥· H(·)+2trace(RN(·,H)·)> = 0

and∆⊥H+ traceσ(·,AH·)+ trace(RN(·,H)·)⊥ = 0,

where ∆⊥ is the Laplacian in the normal bundle.


Introducing the biconservative submanifolds Biconservative hypersurfaces

CorollaryIf M is a hypersurface in a Riemannian manifold N, then M isbiharmonic if and only if

2A(∇f )+ f ∇f −2f (RicciN(η))> = 0

and−∆f − f |A|2 + f RicciN(η ,η) = 0,

where η is the unit normal of M in N and f = traceA is the meancurvature function.


Biconservative surfaces in 3-dimensional space forms


Corollary

A surface M2 in a space form N3(c) is biconservative if and only if

A(∇f ) =− f2

∇f .

Remark

Any CMC surface in N3(c) is biconservative.




Theorem (Caddeo-Montaldo-Oniciuc-Piu, 2014)

Let M2 be a non-CMC biconservative surface in a space form N3(c).There exists an open subset U ⊂M such that, on U, the Gaussiancurvature K of M satisfies

K = detA+ c =−3f 2

4+ c

and(c−K)∆K−|∇K|2− 8

3K(c−K)2 = 0,

where ∆ is the Laplace-Beltrami operator on M.

Remark

For a non-CMC biconservative surface in N3(c) we have c−K > 0.


Biconservative surfaces in 3-dimensional space forms Biconservativity and minimality in space forms

Ricci surfaces

Definition

A Riemannian surface(M2,g

)with Gaussian curvature K is said to

satisfy the Ricci condition if c−K > 0 and the metric (c−K)1/2g is flat,where c ∈R is a constant. In this case,

(M2,g

)is called a Ricci surface.

G. Ricci-Curbastro, 1895, proved that, when c = 0, a surfacesatisfying the Ricci condition can be locally isometricallyembedded in R3 as a minimal surfaceH. B. Lawson, 1970, generalized this result to CMC surfaces inspace forms N3(c)



Ricci surfaces

Proposition (A. Moroianu-S. Moroianu, 2014)

Let(M2,g

)be a Riemannian surface such that its Gaussian curvature

K satisfies c−K > 0, where c ∈ R is a constant. Then, the followingconditions are equivalent:

(i) (c−K)∆K−|∇K|2−4K(c−K)2 = 0;

(ii) ∆ log(c−K)+4K = 0;

(iii) the metric (c−K)1/2g is flat.Moreover, if c = 0, then we also have a fourth equivalent condition:

(iv) the metric (−K)g has constant Gaussian curvature equal to 1.



Proposition (F.-Nistor-Oniciuc, 2015)

Let(M2,g

)be a Riemannian surface such that its Gaussian curvature

K satisfies c−K > 0, where c ∈ R is a constant. Then, the followingconditions are equivalent:

(i) (c−K)∆K−|∇K|2− 83 K(c−K)2 = 0;

(ii) ∆ log(c−K)+(8/3)K = 0;

(iii) the metric (c−K)3/4g is flat.Moreover, if c = 0, then we also have a fourth equivalent condition:

(iv) the metric (−K)g has constant Gaussian curvature equal to 1/3.



Theorem (F.-Nistor-Oniciuc, 2015)

Let(M2,g

)be a Riemannian surface with negative Gaussian curvature

K that satisfiesK∆K + |∇K|2 + 8

3K3 = 0.

Then(M2,(−K)1/2g

)is a Ricci surface in R3.

Corollary

Let(M2,g

)be a biconservative surface in R3, where g is the induced

metric on M. If f (p)> 0 and (∇f )(p) 6= 0 at any point p ∈M, where f isthe mean curvature function, then

(M2,(−K)1/2g

)is a Ricci surface.




Let(M2,g

)be a biconservative surface in a space form N3(c) with

induced metric g and Gaussian curvature K. If f (p)> 0 and (∇f )(p) 6= 0at any point p ∈M, where f is the mean curvature function, then, on anopen dense set,

(M2,(c−K)rg

)is a Ricci surface in N3(c), where r is a

locally defined function that satisfies

K +∆

(14

log(c−Kr)+r2

log(c−K)

)= 0,

with the Gaussian curvature Kr of (c−K)rg given by

Kr = (c−K)−r(

3−4r3

K +12

log(c−K)∆r+(c−K)−1g(∇r,∇K)

).


Biconservative surfaces in 3-dimensional space forms An intrinsic characterization

The characterization theorem


Let (M2,g) be a Riemannian surface and c ∈ R a constant. Then M canbe locally isometrically embedded in a space form N3(c) as abiconservative surface with positive mean curvature having thegradient different from zero at any point p ∈M if and only if theGaussian curvature K satisfies c−K(p)> 0, (∇K)(p) 6= 0, and its levelcurves are circles in M with curvature κ = (3|∇K|)/(8(c−K)).



Theorem (F.-Nistor-Oniciuc, 2015)Let

(M2,g

)be a Riemannian surface with Gaussian curvature K satisfying (∇K)(p) 6= 0 and

c−K(p)> 0 at any point p ∈M, where c ∈ R is a constant. Let X1 = (∇K)/|∇K| and X2 ∈ C(TM)be two vector fields on M such that {X1(p),X2(p)} is a positively oriented orthonormal basis atany point p ∈M. If level curves of K are circles in M with constant curvature

κ =3X1K

8(c−K)=

3|∇K|8(c−K)

,

then, for any point p0 ∈M, there exists a parametrization X = X(u,s) of M in a neighborhoodU ⊂M of p0 positively oriented such that

(a) the curve u→ X(u,0) is an integral curve of X1 with X(0,0) = p0 and s→ X(u,s) is an integralcurve of X2, for any u;

(b) K(u,s) = (K ◦X)(u,s) = (K ◦X)(u,0) = K(u);

(c) g11(u,s) = 964

(K′(u)

c−K(u)

)2s2 +1, g12(u,s) =− 3K′(u)

8(c−K(u)) s, g22(u,s) = 1;

(d) 24(c−K)K′′+33(K′)2 +64K(c−K)2 = 0;

(e) X1 = Xu−g12Xs, X2 = Xs, ∇X1 X1 = ∇X1 X2 = 0, ∇X2 X2 =− 3X1K8(c−K)X1, ∇X2 X1 =

3X1K8(c−K)X2, and,

therefore, the integral curves of X1 are geodesics.



Remark

24(c−K)K′′+33(K′)2 +64K(c−K)2 = 0

⇔

(c−K)∆K−|∇K|2− 83

K(c−K)2 = 0

Remark

(u,s)→(

u,(c−K)3/8s)= (u,v)⇒ g = du2 +(c−K)−

34 dv2

(u,v)→(∫ u

u0

(c−K)3/8du,v)= (u, v)⇒ g = (c−K)−

34(du2 +dv2)



Theorem

Let M2 be a surface and c ∈ R a constant. Consider a fixed pointp0 ∈M, a parametrization X = X(u,s) of M on a neighborhood U ⊂M ofp0 positively oriented, and K = K(u) a function on M such that K′(u)> 0and c−K(u)> 0, for any u, and

24(c−K)K′′+33(K′)2 +64K(c−K)2 = 0.

Define a Riemannian metric g = g11du2 +2g12duds+g22ds2 on U by

g11(u,s) =9

64

(K′(u)

c−K(u)

)2

s2 +1, g12(u,s) =−3K′(u)

8(c−K(u))s, g22(u,s) = 1.

Then K is the Gaussian curvature of g and its level curves, i.e., thecurves s→ X(u,s), are circles in M with curvatureκ = 3K′(u)/(8(c−K(u))).


CMC biconservative surfaces PMC biconservative surfaces in Mn(c)×R

PMC biconservative surfaces in Mn(c)×R

TheoremA submanifold Σm in a Riemannian manifold M is biconservative if andonly if

m2

∇|H|2 +2traceA∇⊥· H(·)+2trace(R(·,H)·)> = 0.



Definition

If the mean curvature vector field H of a surface Σ2 in a Riemannianmanifold M is parallel in the normal bundle, i.e., ∇⊥H = 0, then Σ2 iscalled a PMC surface.

RemarkIn a space of constant curvature, a PMC submanifold is biconservative.

Corollary

Let Σ2 be a non-minimal PMC surface in M = Mn(c)×R, where Mn(c) iseither Sn or Hn. Then Σ2 is biconservative if and only if

〈H,N〉T = 0,

where T and N are the tangent and the normal components of ξ ,respectively.



Theorem (F.-Oniciuc-Pinheiro, 2015)Let Σ2 be a PMC biconservative surface with mean curvature vector field H in M = Mn(c)×R,c =±1 and H 6= 0. Then either

1 Σ2 either is a minimal surface of an umbilical hypersurface of Mn(c) or it is a CMC surfacein a 3-dimensional umbilical submanifold of Mn(c); or

2 Σ2 is a vertical cylinder over a circle in M2(c) with curvature κ = 2|H|; or3 Σ2 lies in S4×R⊂ R5×R and, as a surface in R5×R, is locally given by

X(u,v) =1a{C3 + sinθ(D1 cos(au)+D2 sin(au))}+(ucosθ +b)ξ +

1κ(C1(cosv−1)+C2 sinv),

where θ ∈ (0,π/2) is a constant, a =√

1+ sin2θ , b is a real constant,

κ =

√1+4|H|2 + sin2

θ ,

C1 and C2 are two constant orthonormal vectors in R5×R such that C1 ⊥ ξ and C2 ⊥ ξ , C3is a unit constant vector such that 〈C3,C1〉= a/κ ∈ (0,1), C3 ⊥ C2, and C3 ⊥ ξ , and D1 andD2 are two constant orthonormal vectors in the orthogonal complement ofspan{C1,C2,C3,ξ} in R5×R.



Remark

There are no PMC biconservative surfaces in M3(c)×R, where c =±1,that do not lie in M3(c) nor are vertical cylinders.

RemarkSurfaces given by the third case of the theorem lie in the Riemannianproduct of a small hypersphere of S4 with R.

Remark

It can be proved that PMC biconservative surfaces in M4(c)×R, withc 6= 0 an arbitrary constant, that are not pseudo-umbilical nor verticalcylinders exist only when c > 0.

RemarkSurfaces described in the third case of the theorem are not biharmonic.


CMC biconservative surfaces CMC biconservative surfaces in M3(c)×R

CMC biconservative surfaces in M3(c)×R

Theorem (F.-Oniciuc-Pinheiro, 2015)Let Σ2 be a CMC biconservative surface in M3(c)×R, c 6= 0, with mean curvature vector fieldH 6= 0 orthogonal to ξ and |T| ∈ (0,1). Then Σ2 is flat and it is locally given by X = X(u,v), whereX : D⊂ R2→M3(c)×R is an isometric immersion, D is an open set in R2, and either

1 Σ2 is pseudo-umbilical, c < 0, |H|2 =−c(1−|T|2), the integral curves of Xu are helices suchthat 〈Xu,ξ 〉= |T|, with curvatures κ1

1 = |H| and κ12 =√−c|T|, and the integral curves of Xv

are circles such that 〈Xv,ξ 〉= 0, with curvature κ21 = |H|; or

2 |H|2 >−c(1−|T|2) and the integral curves of Xu and Xv are helices in M3(c)×R satisfying

〈Xu,ξ 〉= a and 〈Xv,ξ 〉= b,

where a,b ∈ R, 0 < a2 +b2 = |T|2 < 1, and |H|2 + c(1−a2−b2)> 0, and with curvatures

κ11 = |H|+

√|H|2 + c(1−a2−b2) and κ

12 =

|a|√1−a2−b2

κ11 ,

κ21 =

∣∣∣|H|−√|H|2 + c(1−a2−b2)∣∣∣ and κ

22 =

|b|√1−a2−b2

κ21 ,

respectively.


CMC biconservative surfaces CMC biconservative surfaces in M3(c)×R

RemarkSurfaces given by the previous theorem, of both pseudo-umbilical andnon-pseudo-umbilical type, do exist.

Theorem (F.-Oniciuc-Pinheiro, 2015)

If Σ2 is a CMC biharmonic surface in M3(c)×R, c 6= 0, with meancurvature vector field H 6= 0 orthogonal to ξ and |T| ∈ (0,1), then c > 0,b2 > a2, and Σ2 is one of the non-pseudo-umbilical CMC biconservativesurfaces in the previous theorem, with

|H|2 = c(1−a2−b2)(b2−a2)2

4(1−a2)(1−b2).


CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds

CMC biconservative surfaces in Hadamard manifolds


Let Σ2 be a non-minimal CMC biconservative surface in a Riemannian manifold M. Then

−12

∆|φH |2 = 2K|φH |2 + |∇φH |2,

where φH = AH −|H|2 Id is the traceless part of AH .

Corollary

Let Σ2 be a CMC biconservative surface in a Riemannian manifold M and assume that Σ2 iscompact and K ≥ 0. Then ∇AH = 0 and Σ2 is pseudo-umbilical or flat.

Corollary

Let Σ2 be a non-minimal CMC biconservative surface in a Riemannian manifold M, with sectionalcurvature bounded from below by a constant K0, such that µ = sup

Σ2 (|σ |2− (1/|H|)2|AH |2)<+∞.Then

∆|φH | ≤ a|φH |3 +b|φH |,

where a and b are constants depending on K0, |H|, and µ.




Let Σ2 be a complete non-minimal CMC biconservative surface in a Hadamard manifold M, withsectional curvature bounded from below by a constant K0 < 0, such that the norm of its secondfundamental form σ is bounded and ∫

Σ2|φH |2 dv <+∞.

Then the function u = |φH | goes to zero uniformly at infinity. More exactly, there exist positiveconstants C0 and C1, depending on K0, |H|, and µ = sup

Σ2 (|σ |2− (1/|H|)2|AH |2), and a positive

radius RΣ2 , determined by C1

∫E(R

Σ2 )u2 dv≤ 1, such that

||u||∞,E(2R) ≤ C0

∫Σ2

u2 dv,

for all R≥ RΣ2 . Moreover, there exist some positive constants D0 and E0, depending on K0, |H|,

and µ, such that the inequality∫

Σ2u2 dv≤ D0 implies

||u||∞ ≤ E0

∫Σ2

u2 dv.



Corollary

Let Σ2 be a complete non-minimal CMC biconservative surface in a 3-dimensional Hadamardmanifold M, with sectional curvature bounded from below by a constant K0 < 0, such that

∫Σ2|φH |2 dv <+∞.

Then the function u = |φH | goes to zero uniformly at infinity. More exactly, there exist positiveconstants C0 and C1, depending on K0 and |H|, and a positive radius R

Σ2 , determined by

C1

∫E(R

Σ2 )u2 dv≤ 1, such that

||u||∞,E(2R) ≤ C0

∫Σ2

u2 dv,

for all R≥ RΣ2 . Moreover, there exist some positive constants D0 and E0, depending on K0 and

|H|, such that the inequality∫

Σ2u2 dv≤ D0 implies

||u||∞ ≤ E0

∫Σ2

u2 dv.




Let Σ2 be a complete non-minimal CMC biconservative surface in aHadamard manifold M, with sectional curvature bounded from belowby a constant K0 < 0, such that the norm of its second fundamentalform σ is bounded, ∫

Σ2|φH|2 dv <+∞,

and |H|2 > (µ−2K0)/2, where µ = supΣ2(|σ |2− (1/|H|2)|AH|2). Then Σ2

is compact.



Corollary

Let Σ2 be a complete non-minimal CMC biconservative surface in a3-dimensional Hadamard manifold M, with sectional curvaturebounded from below by a constant K0 < 0, such that∫

Σ2|φH|2 dv <+∞,

and |H|2 >−K0. Then Σ2 is compact.


References

References

D. Fetcu, S. Nistor, and C. Oniciuc, On biconservative surfacesin 3-dimensional space forms, Comm. Anal. Geom., to appear.

D. Fetcu, C. Oniciuc, and A. L. Pinheiro, CMC biconservativesurfaces in Sn×R and Hn×R, J. Math. Anal. Appl. 425 (2015),588–609.


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