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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
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 2 / 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
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 2 / 30
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)
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 3 / 30
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)
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 3 / 30
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
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 4 / 30
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 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
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 4 / 30
Introducing the biconservative submanifolds History
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 5 / 30
Introducing the biconservative submanifolds History
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 5 / 30
Introducing the biconservative submanifolds History
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 5 / 30
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 6 / 30
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 6 / 30
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 7 / 30
Biconservative surfaces in 3-dimensional space forms
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 8 / 30
Biconservative surfaces in 3-dimensional space forms
Biconservative surfaces in 3-dimensional space forms
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 9 / 30
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)
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 10 / 30
Biconservative surfaces in 3-dimensional space forms Biconservativity and minimality in space forms
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 11 / 30
Biconservative surfaces in 3-dimensional space forms Biconservativity and minimality in space forms
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 12 / 30
Biconservative surfaces in 3-dimensional space forms Biconservativity and minimality in space forms
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 13 / 30
Biconservative surfaces in 3-dimensional space forms Biconservativity and minimality in space forms
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 13 / 30
Biconservative surfaces in 3-dimensional space forms Biconservativity and minimality in space forms
Theorem (F.-Nistor-Oniciuc, 2015)
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)
).
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 14 / 30
Biconservative surfaces in 3-dimensional space forms An intrinsic characterization
The characterization theorem
Theorem (F.-Nistor-Oniciuc, 2015)
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)).
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 15 / 30
Biconservative surfaces in 3-dimensional space forms An intrinsic characterization
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 16 / 30
Biconservative surfaces in 3-dimensional space forms An intrinsic characterization
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)
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 17 / 30
Biconservative surfaces in 3-dimensional space forms An intrinsic characterization
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)
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 17 / 30
Biconservative surfaces in 3-dimensional space forms An intrinsic characterization
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))).
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 18 / 30
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 19 / 30
CMC biconservative surfaces PMC biconservative surfaces in Mn(c)×R
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 20 / 30
CMC biconservative surfaces PMC biconservative surfaces in Mn(c)×R
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 20 / 30
CMC biconservative surfaces PMC biconservative surfaces in Mn(c)×R
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 20 / 30
CMC biconservative surfaces PMC biconservative surfaces in Mn(c)×R
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 21 / 30
CMC biconservative surfaces PMC biconservative surfaces in Mn(c)×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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 22 / 30
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 23 / 30
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).
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 24 / 30
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).
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 24 / 30
CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds
CMC biconservative surfaces in Hadamard manifolds
Theorem (F.-Oniciuc-Pinheiro, 2015)
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 25 / 30
CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds
CMC biconservative surfaces in Hadamard manifolds
Theorem (F.-Oniciuc-Pinheiro, 2015)
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 25 / 30
CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds
CMC biconservative surfaces in Hadamard manifolds
Theorem (F.-Oniciuc-Pinheiro, 2015)
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 25 / 30
CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds
Theorem (F.-Oniciuc-Pinheiro, 2015)
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 26 / 30
CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 27 / 30
CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds
Theorem (F.-Oniciuc-Pinheiro, 2015)
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 28 / 30
CMC biconservative surfaces CMC biconservative surfaces in Hadamard manifolds
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 29 / 30
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.
Dorel Fetcu (TUIASI) On biconservative surfaces Brno, Czechia, July 2016 30 / 30