# On biconservative Biconservative surfaces in 3-dimensional space forms Biconservative surfaces in...

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On biconservative surfaces

Dorel Fetcu

Gheorghe Asachi Technical University of Iaşi, 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) = 1 2

∫ M |dϕ|2vg

Euler-Lagrange equation

τ(ϕ) = τ1(ϕ) = traceg∇dϕ = 0

Critical points of E: harmonic maps

Bienergy functional

E2 (ϕ) = 1 2

∫ 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) = 1 2

∫ M |dϕ|2vg

Euler-Lagrange equation

τ(ϕ) = τ1(ϕ) = traceg∇dϕ = 0

Critical points of E: harmonic maps

Bienergy functional

E2 (ϕ) = 1 2

∫ 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) =−∆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 = ∇NX ∇ N Y Z−∇NY ∇NX Z−∇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 a submanifold M of a Riemannian manifold N is called a biharmonic 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) =−∆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 = ∇NX ∇ N Y Z−∇NY ∇NX Z−∇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 a submanifold M of a Riemannian manifold N is called a biharmonic 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 the stress-energy tensor

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

S = 1 2 |dϕ|2g−ϕ∗h

that satisfies divS =−〈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 the stress-energy tensor P. Baird and J. Eells, 1981; A. Sanini, 1983, used the tensor

S = 1 2 |dϕ|2g−ϕ∗h

that satisfies divS =−〈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 the bienergy:

S2(X,Y) = 1 2 |τ(ϕ)|2〈X,Y〉+ 〈dϕ,∇τ(ϕ)〉〈X,Y〉

−〈dϕ(X),∇Yτ(ϕ)〉−〈dϕ(Y),∇Xτ(ϕ)〉

that satisfies divS2 =−〈τ2(ϕ),dϕ〉

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

Remark If ϕ : M→ (N,h) is a fixed map, then E2 can be thought as a functional on the set of all Riemannian metrics on M. This new functional’s critical 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 the bienergy:

S2(X,Y) = 1 2 |τ(ϕ)|2〈X,Y〉+ 〈dϕ,∇τ(ϕ)〉〈X,Y〉

−〈dϕ(X),∇Yτ(ϕ)〉−〈dϕ(Y),∇Xτ(ϕ)〉

that satisfies divS2 =−〈τ2(ϕ),dϕ〉

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

Remark If ϕ : M→ (N,h) is a fixed map, then E2 can be thought as a functional on the set of all Riemannian metrics on M. This new functional’s critical 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 the bienergy:

S2(X,Y) = 1 2 |τ(ϕ)|2〈X,Y〉+ 〈dϕ,∇τ(ϕ)〉〈X,Y〉

−〈dϕ(X),∇Yτ(ϕ)〉−〈dϕ(Y),∇Xτ(ϕ)〉

that satisfies divS2 =−〈τ2(ϕ),dϕ〉

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

Remark If ϕ : M→ (N,h) is a fixed map, then E2 can be thought as a functional on the set of all Riemannian metrics on M. This new functional’s critical 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

Definition A submanifold ϕ : M→ N of a Riemannian manifold N is called a biconservative submanifold if divS2 = 0, i.e., τ2(ϕ)> = 0.

Theorem (Balmuş-Montaldo-Oniciuc, 2012)

A submanifold M in a Riemannian manifold N, with second fundamental form σ , mean curvature vector field H, and shape operator A, is biharmonic if and only if

m 2

∇|H|2 +2traceA∇⊥· H(·)+2trace(R N(·,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

Definition A submanifold ϕ : M→ N of a Riemannian manifold N is called a biconservative submanifold if divS2 = 0, i.e., τ2(ϕ)> = 0.

Theorem (Balmuş-Montaldo-Oniciuc, 2012)

A submanifold M in a Riemannian manifold N, with second fundamental form σ , mean curvature vector field H, and shape operator A, is biharmonic if and only if

m 2

∇|H|2 +2traceA∇⊥· H(·)+2trace(R N(·,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

Corollary If M is a hypersurface in a Riemannian manifold N, then M is biharmonic 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 mean curvature 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 ) =− f 2

∇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 Gaussian curvature K of M satisfies

K = detA+ c =−3f 2

4 + c

and (c−K)∆K−|∇K|2− 8

3 K(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 spa

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