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On the geometry of biharmonicmaps and biharmonic
submanifolds
Adina Balmus, Stefano Montaldo and Cezar Oniciuc
Universita degli Studi di Cagliari
“Al.I. Cuza” University of Iasi
Constanta - August 2011
1 / 37
Chen definitionLet
i : M → Rn
be the canonical inclusion and H = (H1, . . . ,Hn) the mean cur-vature vector field.
Definition (B-Y. Chen) A submanifold M ⊂ Rn is biharmonic iff
∆H = (∆H1, . . . ,∆Hn) = 0
where ∆ is the Beltrami-Laplace operator on M w.r.t. the metricinduced by i.
2 / 37
Chen definitionLet
i : M → Rn
be the canonical inclusion and H = (H1, . . . ,Hn) the mean cur-vature vector field.
Definition (B-Y. Chen) A submanifold M ⊂ Rn is biharmonic iff
∆H = (∆H1, . . . ,∆Hn) = 0
where ∆ is the Beltrami-Laplace operator on M w.r.t. the metricinduced by i.
• Why biharmonic?
m∆H = ∆(−∆i) = −∆2i
2 / 37
Chen definitionLet
i : M → Rn
be the canonical inclusion and H = (H1, . . . ,Hn) the mean cur-vature vector field.
Definition (B-Y. Chen) A submanifold M ⊂ Rn is biharmonic iff
∆H = (∆H1, . . . ,∆Hn) = 0
where ∆ is the Beltrami-Laplace operator on M w.r.t. the metricinduced by i.
• Why biharmonic?
m∆H = ∆(−∆i) = −∆2i
• CMC submanifolds, |H| = constant, are not necessarily bi-harmonic.
2 / 37
Biharmonic submanifolds in En(c)
Leti : Mm → E
n(c)
be the canonical inclusion of a submanifold M in a constant sec-tional curvature c manifold.
3 / 37
Biharmonic submanifolds in En(c)
Leti : Mm → E
n(c)
be the canonical inclusion of a submanifold M in a constant sec-tional curvature c manifold.
Definition M is a biharmonic submanifold iff
∆iH = mcH
where• H ∈ C(i−1(TEn(c))) denotes the mean curvature vector fieldof M in E
n(c)
• ∆i is the rough Laplacian on i−1(TEn(c))
3 / 37
RemarkIf En(c) = S
n then one can consider Sn ⊂ Rn+1 and the inclusion
i : Mm → Sn ⊂ R
n+1
can be seen as a map into Rn+1.
4 / 37
RemarkIf En(c) = S
n then one can consider Sn ⊂ Rn+1 and the inclusion
i : Mm → Sn ⊂ R
n+1
can be seen as a map into Rn+1.
Alternative problem (Alias, Barros, Ferrandez)
∆H′ = (∆H1, . . . ,∆Hn+1) = λH′
where H′ is the mean curvature vector field of the inclusion as a
map into Rn+1.
4 / 37
RemarkIf En(c) = S
n then one can consider Sn ⊂ Rn+1 and the inclusion
i : Mm → Sn ⊂ R
n+1
can be seen as a map into Rn+1.
Alternative problem (Alias, Barros, Ferrandez)
∆H′ = (∆H1, . . . ,∆Hn+1) = λH′
where H′ is the mean curvature vector field of the inclusion as a
map into Rn+1.
This is NOT the biharmonic condition for submanifolds in Sn
∆H′ = mcH′
��HH⇔ ∆iH = mcH
4 / 37
Where does this definition come from?
To understand the origin of the biharmonic equation we need touse the theory of harmonic maps.
5 / 37
The energy Functional
Harmonic maps ϕ : (M,g) → (N,h) are critical points of theenergy
E (ϕ) =1
2
∫
M
|dϕ|2 vg
and they are solutions of the Euler-Lagrange equation
τ(ϕ) = traceg∇dϕ = 0
6 / 37
The energy Functional
Harmonic maps ϕ : (M,g) → (N,h) are critical points of theenergy
E (ϕ) =1
2
∫
M
|dϕ|2 vg
and they are solutions of the Euler-Lagrange equation
τ(ϕ) = traceg∇dϕ = 0
• If ϕ is an isometric immersion, with mean curvature vectorfield H, then:
τ(ϕ) = mH
6 / 37
The bienergy Functional
The bienergy functional (proposed by Eells–Lemaire) is
E2 (ϕ) =1
2
∫
M
|τ(ϕ)|2 vg
Critical points of E2 are called biharmonic maps and they aresolutions of the Euler-Lagrange equation (Jiang):
τ2(ϕ) = −∆ϕτ(ϕ) − traceg RN (dϕ, τ(ϕ))dϕ = 0
where RN is the curvature operator on N .
7 / 37
The bienergy Functional
The bienergy functional (proposed by Eells–Lemaire) is
E2 (ϕ) =1
2
∫
M
|τ(ϕ)|2 vg
Critical points of E2 are called biharmonic maps and they aresolutions of the Euler-Lagrange equation (Jiang):
τ2(ϕ) = −∆ϕτ(ϕ) − traceg RN (dϕ, τ(ϕ))dϕ = 0
where RN is the curvature operator on N .
• The biharmonic equation
τ2(ϕ) = 0
is a fourth-order non-linear elliptic equation (not easy to solve!).
7 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
Proof
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
Proof ϕ biharmonic ⇒
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
Proof ϕ biharmonic ⇒ ∆ϕτ = − traceRN (dϕ, τ)dϕ
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
Proof ϕ biharmonic ⇒ ∆ϕτ = − traceRN (dϕ, τ)dϕ
∫
M
|∇τ |2 vg =∫
M
〈∆ϕτ, τ〉 vg = − trace
∫
M
〈RN (dϕ, τ)dϕ, τ〉 vg ≤ 0
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
Proof ϕ biharmonic ⇒ ∆ϕτ = − traceRN (dϕ, τ)dϕ
∫
M
|∇τ |2 vg =∫
M
〈∆ϕτ, τ〉 vg = − trace
∫
M
〈RN (dϕ, τ)dϕ, τ〉 vg ≤ 0
⇒ ∇τ = 0.
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
Proof ϕ biharmonic ⇒ ∆ϕτ = − traceRN (dϕ, τ)dϕ
∫
M
|∇τ |2 vg =∫
M
〈∆ϕτ, τ〉 vg = − trace
∫
M
〈RN (dϕ, τ)dϕ, τ〉 vg ≤ 0
⇒ ∇τ = 0. From
div〈τ, dϕ〉 = trace〈∇τ, dϕ〉 + |τ |2
8 / 37
Non existence of Biharmonic Maps
Harmonic maps are trivially biharmonicA non harmonic biharmonic map is called proper biharmonic
Proposition (Jiang) If M is compact and SecN ≤ 0 then bihar-monic ⇒ harmonic
Proof ϕ biharmonic ⇒ ∆ϕτ = − traceRN (dϕ, τ)dϕ
∫
M
|∇τ |2 vg =∫
M
〈∆ϕτ, τ〉 vg = − trace
∫
M
〈RN (dϕ, τ)dϕ, τ〉 vg ≤ 0
⇒ ∇τ = 0. From
0 =
∫
M
div〈τ, dϕ〉 vg =
∫
M
|τ |2 vg
⇒ τ = 0. Q.E.D.
8 / 37
Remarks
• M compact + SecN ≤ 0 ⇒ there exists a harmonic map
ϕ : M → N
in each homotopy class (Eells–Sampson)
9 / 37
Remarks
• M compact + SecN ≤ 0 ⇒ there exists a harmonic map
ϕ : M → N
in each homotopy class (Eells–Sampson)
• Harmonic maps do not always exists. There exists no har-monic map from
T2 → S
2
in the homotopy class of Brower degree ±1 (Eells–Wood)
9 / 37
Remarks
• M compact + SecN ≤ 0 ⇒ there exists a harmonic map
ϕ : M → N
in each homotopy class (Eells–Sampson)
• Harmonic maps do not always exists. There exists no har-monic map from
T2 → S
2
in the homotopy class of Brower degree ±1 (Eells–Wood)
Problem Find biharmonic maps T2 → S
2 of degree ±1
9 / 37
Remarks
• M compact + SecN ≤ 0 ⇒ there exists a harmonic map
ϕ : M → N
in each homotopy class (Eells–Sampson)
• Harmonic maps do not always exists. There exists no har-monic map from
T2 → S
2
in the homotopy class of Brower degree ±1 (Eells–Wood)
Problem Find biharmonic maps T2 → S
2 of degree ±1
• So far we only know examples of biharmonic maps T2 → S
2
whose image is a curve.
9 / 37
Examples of proper biharmonic maps
10 / 37
Examples of proper biharmonic maps
• Any polynomial map of degree 3 between Euclidean spaces
10 / 37
Examples of proper biharmonic maps
• Any polynomial map of degree 3 between Euclidean spaces
Property (Almansi): let f : Rn → R be any harmonic functionthen
g(x) = |x|ℓf(x)is proper biharmonic for any harmonic f if and only if ℓ = 2.
10 / 37
Examples of proper biharmonic maps
• Any polynomial map of degree 3 between Euclidean spaces
Property (Almansi): let f : Rn → R be any harmonic functionthen
g(x) = |x|ℓf(x)is proper biharmonic for any harmonic f if and only if ℓ = 2.
• From the Hopf map H : C2 → R × C we get the proper
biharmonic map
C2 → R× C, (z, w) 7→ (|z|2 + |w|2)(|z|2 − |w|2, 2zw)
10 / 37
Examples of proper biharmonic maps
• Any polynomial map of degree 3 between Euclidean spaces
Property (Almansi): let f : Rn → R be any harmonic functionthen
g(x) = |x|ℓf(x)is proper biharmonic for any harmonic f if and only if ℓ = 2.
• From the Hopf map H : C2 → R × C we get the proper
biharmonic map
C2 → R× C, (z, w) 7→ (|z|2 + |w|2)(|z|2 − |w|2, 2zw)
• Let f(x1, ..., xn) =∑n
i=1aixi, ai ∈ R, then
g(x) = |x|2−nf(x)
is proper biharmonic (M–Impera)10 / 37
Examples of proper biharmonic maps
• The generalized Kelvin transformation
ϕ : Rm \ {0} → Rm \ {0}, ϕ(p) =
p
|p|ℓ
is proper biharmonic iff ℓ = m− 2 (B–M–O)
11 / 37
Examples of proper biharmonic maps
• The generalized Kelvin transformation
ϕ : Rm \ {0} → Rm \ {0}, ϕ(p) =
p
|p|ℓ
is proper biharmonic iff ℓ = m− 2 (B–M–O)
• The quaternionic multiplication
H → H, q 7→ qn
is biharmonic for any n ∈ N (Fueter, 1935)
11 / 37
Lets go back to biharmonic submanifolds
If ϕ : M → En(c) is an isometric immersion then
τ(ϕ) = mH, τ2(ϕ) = −m∆ϕH+ cm2
H
thus ϕ is biharmonic iff
∆ϕH = mcH
12 / 37
Lets go back to biharmonic submanifolds
If ϕ : M → En(c) is an isometric immersion then
τ(ϕ) = mH, τ2(ϕ) = −m∆ϕH+ cm2
H
thus ϕ is biharmonic iff
∆ϕH = mcH
Moreover, if ϕ : M → Rn is anisometric immersion, set ϕ =
(ϕ1, . . . , ϕn) and H = (H1, . . . ,Hn). Then
∆ϕH = (∆H1, . . . ,∆Hn)
and we recover Chen’s definition.
12 / 37
Biharmonic submanifolds of En(c)
13 / 37
Biharmonic submanifolds of En(c)
Proposition [Chen (c = 0) - Oniciuc (c ≤ 0)] Let
ϕ : M → En(c)
be an isometric immersion with |H| = constant. If c ≤ 0, then ϕis biharmonic iff H = 0.
13 / 37
Biharmonic submanifolds of En(c)
Proposition [Chen (c = 0) - Oniciuc (c ≤ 0)] Let
ϕ : M → En(c)
be an isometric immersion with |H| = constant. If c ≤ 0, then ϕis biharmonic iff H = 0.
Proof Biharmonic ⇒ ∆ϕH = mcH.
13 / 37
Biharmonic submanifolds of En(c)
Proposition [Chen (c = 0) - Oniciuc (c ≤ 0)] Let
ϕ : M → En(c)
be an isometric immersion with |H| = constant. If c ≤ 0, then ϕis biharmonic iff H = 0.
Proof Biharmonic ⇒ ∆ϕH = mcH. Replacing in the Weitzen-
bock formula1
2∆ϕ|H|2 = 〈∆ϕ
H,H〉 − |∇H|2
we get, since c ≤ 0,
|∇H|2 = mc |H|2 ≤ 0
Thus we conclude that ∇H = 0.
13 / 37
Biharmonic submanifolds of En(c)
Proposition [Chen (c = 0) - Oniciuc (c ≤ 0)] Let
ϕ : M → En(c)
be an isometric immersion with |H| = constant. If c ≤ 0, then ϕis biharmonic iff H = 0.
Proof Biharmonic ⇒ ∆ϕH = mcH. Replacing in the Weitzen-
bock formula1
2∆ϕ|H|2 = 〈∆ϕ
H,H〉 − |∇H|2
we get, since c ≤ 0,
|∇H|2 = mc |H|2 ≤ 0
Thus we conclude that ∇H = 0.Next, for an isometric immersion we have:
|H|2 = − trace〈dϕ,∇H〉Q.E.D.
13 / 37
Geometric conditions for biharmonic submanifoldsBy decomposing the equation ∆ϕ
H = mcH in its normal andtangent components we find that an isometric immersion ϕ :Mm → E
n(c) is biharmonic iff
−∆⊥H− traceB(·, AH·) +mcH = 0 (normal)
2 traceA∇⊥
(·)H(·) + m
2grad(|H|2) = 0 (tangent)
A is the Weingarten operator - B the second fundamental form∇⊥ and ∆⊥ the connection and the Laplacian in the normal bun-dle.
14 / 37
Geometric conditions for biharmonic submanifoldsBy decomposing the equation ∆ϕ
H = mcH in its normal andtangent components we find that an isometric immersion ϕ :Mm → E
n(c) is biharmonic iff
−∆⊥H− traceB(·, AH·) +mcH = 0 (normal)
2 traceA∇⊥
(·)H(·) + m
2grad(|H|2) = 0 (tangent)
A is the Weingarten operator - B the second fundamental form∇⊥ and ∆⊥ the connection and the Laplacian in the normal bun-dle.
For hypersurfaces
∆⊥H− (mc− |A|2)H = 0
2A(
grad(|H|))
+m|H| grad(|H|) = 0
14 / 37
Non existence of biharmonic submanifolds
15 / 37
Non existence of biharmonic submanifolds
Proposition [Chen (c = 0), Caddeo–M–O (c ≤ 0)] If c ≤ 0, thereexists no proper biharmonic surfaces M2 ⊂ E
3(c).
15 / 37
Non existence of biharmonic submanifolds
Proposition [Chen (c = 0), Caddeo–M–O (c ≤ 0)] If c ≤ 0, thereexists no proper biharmonic surfaces M2 ⊂ E
3(c).
Proof
15 / 37
Non existence of biharmonic submanifolds
Proposition [Chen (c = 0), Caddeo–M–O (c ≤ 0)] If c ≤ 0, thereexists no proper biharmonic surfaces M2 ⊂ E
3(c).
Proof In this caseH = |H|η,
where η is a unit normal to M .
15 / 37
Non existence of biharmonic submanifolds
Proposition [Chen (c = 0), Caddeo–M–O (c ≤ 0)] If c ≤ 0, thereexists no proper biharmonic surfaces M2 ⊂ E
3(c).
Proof In this caseH = |H|η,
where η is a unit normal to M .Then, biharmonicity, Gauss and Codazzi equations imply that|H| is a solution of a polynomial equation with constant coeffi-cients, thus |H| is constant.
Q.E.D.
15 / 37
Chen’s Conjecture
Conjecture
Biharmonic submanifolds of En(c), n > 3, c ≤ 0, are minimal
16 / 37
Chen’s Conjecture
Conjecture
Biharmonic submanifolds of En(c), n > 3, c ≤ 0, are minimal
Partial solutions of the conjecture are known for:
• curves of Rn (Dimitric)• submanifolds of finite type in R
n (Dimitric)• hypersurfaces with at most two principal curvatures (B–M–O)• pseudo-umbilical submanifolds Mm ⊂ E
n(c), c ≤ 0, m 6= 4,(Caddeo–M–O, Dimitric)
• hypersurfaces of E4(c), c ≤ 0 (Hasanis–Vlachos, B–M–O)• spherical submanifolds of Rn (Chen)• submanifolds of bounded geometry (Ichiyama–Inoguchi–Urakawa)
16 / 37
Biharmonic submanifolds of Sn
All the non existence results described in the previous sectiondo not hold for submanifolds in the sphere.
Problem:Classify all biharmonic submanifolds of Sn
17 / 37
Main examples of biharmonic submanifolds in Sn
18 / 37
Main examples of biharmonic submanifolds in Sn
B1. The small hypersphere
Sm( 1√
2) S
m+1biharmonic
18 / 37
Main examples of biharmonic submanifolds in Sn
B1. The small hypersphere
Sm( 1√
2) S
m+1biharmonic
B2. The standard products of spheres
Sm1( 1√
2)× S
m2( 1√2) S
m+1biharmonic
m1 +m2 = m− 1 and m1 6= m2.
18 / 37
Main examples of biharmonic submanifolds in Sn
B3. Composition property
MmSn−1( 1√
2) S
nminimal biharmonic
proper biharmonic
19 / 37
Main examples of biharmonic submanifolds in Sn
B3. Composition property
MmSn−1( 1√
2) S
nminimal biharmonic
proper biharmonic
B4. Product composition property
Mm11
×Mm22
Sn1( 1√
2)× S
n2( 1√2) S
nminimal
proper biharmonic
n1 + n2 = n− 1, m1 6= m2
19 / 37
Biharmonic cuves in Sn
(Caddeo–M–O)
20 / 37
Biharmonic cuves in Sn
(Caddeo–M–O)
γ ⊂ Sn biharmonic curve
κ = curvature
20 / 37
Biharmonic cuves in Sn
(Caddeo–M–O)
γ ⊂ Sn biharmonic curve
κ = curvature
κ = 1
20 / 37
Biharmonic cuves in Sn
(Caddeo–M–O)
γ ⊂ Sn biharmonic curve
κ = curvature
κ = 1
S1( 1√
2) S
2
20 / 37
Biharmonic cuves in Sn
(Caddeo–M–O)
γ ⊂ Sn biharmonic curve
κ = curvature
κ = 1
S1( 1√
2) S
2
κ ∈ (0, 1)
20 / 37
Biharmonic cuves in Sn
(Caddeo–M–O)
γ ⊂ Sn biharmonic curve
κ = curvature
κ = 1
S1( 1√
2) S
2
κ ∈ (0, 1)
γ S1( 1√
2)× S
1( 1√2) S
3
geo
biharmonic
γ geodesic of slope 6= ±1, 0 and ∞
20 / 37
Biharmonic hypersurfaces in Sn
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
κ ≤ 2
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
κ ≤ 2
Sm1( 1√
2)× S
m2( 1√2) S
m+1
B2
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
κ ≤ 2
Sm1( 1√
2)× S
m2( 1√2) S
m+1
B2
κ = 3
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
κ ≤ 2
Sm1( 1√
2)× S
m2( 1√2) S
m+1
B2
κ = 3 Compact + CMC
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
κ ≤ 2
Sm1( 1√
2)× S
m2( 1√2) S
m+1
B2
κ = 3 Compact + CMC
Non Existence
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
κ ≤ 2
Sm1( 1√
2)× S
m2( 1√2) S
m+1
B2
κ = 3 Compact + CMC
Non Existence
Isoparametric
21 / 37
Biharmonic hypersurfaces in Sn
Mm ⊂ Sm+1 biharmonic
κ = number of distinct principal curvature
κ = 1
Sm( 1√
2) S
m+1
B1
κ ≤ 2
Sm1( 1√
2)× S
m2( 1√2) S
m+1
B2
κ = 3 Compact + CMC
Non Existence
Isoparametric
Ichiyama-Inoguchi-Urakawa
21 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMC
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact |H| = 1 |H| ∈ (0, 1)
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact |H| = 1 |H| ∈ (0, 1)
MmSn−1( 1√
2) S
nminimal biharmonic
B3
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact |H| = 1 |H| ∈ (0, 1)
MmSn−1( 1√
2) S
nminimal biharmonic
B3
Compact
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact |H| = 1 |H| ∈ (0, 1)
MmSn−1( 1√
2) S
nminimal biharmonic
B3
Compact
M is of 2-type
22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact |H| = 1 |H| ∈ (0, 1)
MmSn−1( 1√
2) S
nminimal biharmonic
B3
Compact
M is of 2-type
M is of 1-type22 / 37
CMC Biharmonic submanifolds in Sn
Mm ⊂ Sn biharmonic + CMCM3 ⊂ S
4
Compact |H| = 1 |H| ∈ (0, 1)
MmSn−1( 1√
2) S
nminimal biharmonic
B3
Compact
M is of 2-type
M is of 1-type
?
22 / 37
Biharmonic submanifolds in Sn with ∇⊥H = 0
Mm ⊂ Sn biharmonic + PMC
23 / 37
Biharmonic submanifolds in Sn with ∇⊥H = 0
Mm ⊂ Sn biharmonic + PMC
M2 ⊂ Sn
23 / 37
Biharmonic submanifolds in Sn with ∇⊥H = 0
Mm ⊂ Sn biharmonic + PMC
M2 ⊂ Sn
M2Sn−1( 1√
2) S
nmin
B3
23 / 37
Biharmonic submanifolds in Sn with ∇⊥H = 0
Mm ⊂ Sn biharmonic + PMC
M2 ⊂ Sn
M2Sn−1( 1√
2) S
nmin
B3
∇⊥B = 0
23 / 37
Biharmonic submanifolds in Sn with ∇⊥H = 0
Mm ⊂ Sn biharmonic + PMC
M2 ⊂ Sn
M2Sn−1( 1√
2) S
nmin
B3
∇⊥B = 0
Mm11
×Mm22
Sn1( 1√
2)× S
n2( 1√2) S
nmin
B4
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Pseudo-umbilical biharmonic submanifolds in Sn
CMC proper biharmonic submanifolds with |H| = 1 in Sn are B3
and they are pseudo-umbilical:
AH = |H|2Id
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Pseudo-umbilical biharmonic submanifolds in Sn
CMC proper biharmonic submanifolds with |H| = 1 in Sn are B3
and they are pseudo-umbilical:
AH = |H|2Id
Question When a proper biharmonic pseudo-umbilical sub-manifold in S
n has |H| = 1, thus B3?
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Pseudo-umbilical biharmonic submanifolds in Sn
CMC proper biharmonic submanifolds with |H| = 1 in Sn are B3
and they are pseudo-umbilical:
AH = |H|2Id
Question When a proper biharmonic pseudo-umbilical sub-manifold in S
n has |H| = 1, thus B3?
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
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Pseudo-umbilical biharmonic submanifolds in Sn
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
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Pseudo-umbilical biharmonic submanifolds in Sn
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
Proof The tangent part of τ2 becomes
(m− 4) grad |H|2 = 0,
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Pseudo-umbilical biharmonic submanifolds in Sn
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
Proof The tangent part of τ2 becomes
(m− 4) grad |H|2 = 0,
m 6= 4 ⇒ |H| = constant.
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Pseudo-umbilical biharmonic submanifolds in Sn
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
Proof The tangent part of τ2 becomes
(m− 4) grad |H|2 = 0,
m 6= 4 ⇒ |H| = constant.
compact + CMC + pseudo-umbilical ⇒ PMC (H. Li)
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Pseudo-umbilical biharmonic submanifolds in Sn
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
Proof The tangent part of τ2 becomes
(m− 4) grad |H|2 = 0,
m 6= 4 ⇒ |H| = constant.
compact + CMC + pseudo-umbilical ⇒ PMC (H. Li)
⇒ M is minimal in Sn−1(a) ⊂ S
n, a ∈ (0, 1) (Chen)
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Pseudo-umbilical biharmonic submanifolds in Sn
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
Proof The tangent part of τ2 becomes
(m− 4) grad |H|2 = 0,
m 6= 4 ⇒ |H| = constant.
compact + CMC + pseudo-umbilical ⇒ PMC (H. Li)
⇒ M is minimal in Sn−1(a) ⊂ S
n, a ∈ (0, 1) (Chen)
biharmonicity ⇒ a = 1/√2 (Caddeo–M–O)
25 / 37
Pseudo-umbilical biharmonic submanifolds in Sn
Theorem Let Mm be a compact pseudo-umbilical submani-fold in S
n, m 6= 4. Then M is proper biharmonic if and only if Mis B3.
Proof The tangent part of τ2 becomes
(m− 4) grad |H|2 = 0,
m 6= 4 ⇒ |H| = constant.
�����XXXXXcompact + CMC + pseudo-umbilical ⇒ PMC
⇒ M is minimal in Sn−1(a) ⊂ S
n, a ∈ (0, 1) (Chen)
biharmonicity ⇒ a = 1/√2 (Caddeo–M–O)
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The examples of Sasahara et alTheorem Let ϕ : M3 → S
5 be a proper biharmonic anti-invariantimmersion. Then the position vector field x0 = x0(u, v, w) in R
6
is given by
x0(u, v, w) = eiw(eiu, ie−iu sin√2v, ie−iu cos
√2v)
Moreover, |H| = 1/3.
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The examples of Sasahara et alTheorem Let ϕ : M3 → S
5 be a proper biharmonic anti-invariantimmersion. Then the position vector field x0 = x0(u, v, w) in R
6
is given by
x0(u, v, w) = eiw(eiu, ie−iu sin√2v, ie−iu cos
√2v)
Moreover, |H| = 1/3.
The immersion ϕ is PMC but NOT pseudo-umbilical
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The examples of Sasahara et alTheorem Let ϕ : M3 → S
5 be a proper biharmonic anti-invariantimmersion. Then the position vector field x0 = x0(u, v, w) in R
6
is given by
x0(u, v, w) = eiw(eiu, ie−iu sin√2v, ie−iu cos
√2v)
Moreover, |H| = 1/3.
The immersion ϕ is PMC but NOT pseudo-umbilical
Theorem Let φ : M2 → S5 be a proper biharmonic Legendre
immersion. Then the position vector field x0 = x0(u, v) of M inR6 is given by:
x0(u, v) =1√2
(
cos u, sin u sin(√2v),− sin u cos(
√2v),
sinu, cos u sin(√2v),− cos u cos(
√2v)
)
.
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The examples of Sasahara et alTheorem Let ϕ : M3 → S
5 be a proper biharmonic anti-invariantimmersion. Then the position vector field x0 = x0(u, v, w) in R
6
is given by
x0(u, v, w) = eiw(eiu, ie−iu sin√2v, ie−iu cos
√2v)
Moreover, |H| = 1/3.
The immersion ϕ is PMC but NOT pseudo-umbilical
Theorem Let φ : M2 → S5 be a proper biharmonic Legendre
immersion. Then the position vector field x0 = x0(u, v) of M inR6 is given by:
x0(u, v) =1√2
(
cos u, sin u sin(√2v),− sin u cos(
√2v),
sinu, cos u sin(√2v),− cos u cos(
√2v)
)
.
The immersion φ is NOT PMC26 / 37
Open Problems
Conjecture
The only proper biharmonic hypersurfaces in Sn are B1 or B2.
Conjecture
Any biharmonic submanifold in Sn has constant mean curvature.
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Remark
An isometric immersion ϕ : Mm → En(c) is biharmonic iff
−∆⊥H− traceB(·, AH·) +mcH = 0 normal
2 traceA∇⊥
(·)H(·) + m
2grad(|H|2) = 0 tangent
Most of the classification results described depend only on thetangent part of τ2.
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Remark
An isometric immersion ϕ : Mm → En(c) is biharmonic iff
−∆⊥H− traceB(·, AH·) +mcH = 0 normal
2 traceA∇⊥
(·)H(·) + m
2grad(|H|2) = 0 tangent
Most of the classification results described depend only on thetangent part of τ2.
Has the conditionτ2(ϕ)
⊤ = 0
a variational meaning?
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Remark
An isometric immersion ϕ : Mm → En(c) is biharmonic iff
−∆⊥H− traceB(·, AH·) +mcH = 0 normal
2 traceA∇⊥
(·)H(·) + m
2grad(|H|2) = 0 tangent
Most of the classification results described depend only on thetangent part of τ2.
Has the conditionτ2(ϕ)
⊤ = 0
a variational meaning?
YES
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The stress-energy tensorAs described by Hilbert, the stress-energy tensor associated toa variational problem is a symmetric 2-covariant tensor field Sconservative at critical points, i.e. divS = 0 at these points.
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The stress-energy tensorAs described by Hilbert, the stress-energy tensor associated toa variational problem is a symmetric 2-covariant tensor field Sconservative at critical points, i.e. divS = 0 at these points.
• In the context of harmonic maps, the stress-energy tensor is
S =1
2|dϕ|2g − ϕ∗h,
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The stress-energy tensorAs described by Hilbert, the stress-energy tensor associated toa variational problem is a symmetric 2-covariant tensor field Sconservative at critical points, i.e. divS = 0 at these points.
• In the context of harmonic maps, the stress-energy tensor is
S =1
2|dϕ|2g − ϕ∗h, divS = −〈τ(ϕ), dϕ〉
(Baird–Eells)
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The stress-energy tensorAs described by Hilbert, the stress-energy tensor associated toa variational problem is a symmetric 2-covariant tensor field Sconservative at critical points, i.e. divS = 0 at these points.
• In the context of harmonic maps, the stress-energy tensor is
S =1
2|dϕ|2g − ϕ∗h, divS = −〈τ(ϕ), dϕ〉
(Baird–Eells)
• For biharmonic maps the stress-energy tensor is
S2(X,Y ) =1
2|τ(ϕ)|2〈X,Y 〉+ 〈dϕ,∇τ(ϕ)〉〈X,Y 〉
−〈dϕ(X),∇Y τ(ϕ)〉 − 〈dϕ(Y ),∇Xτ(ϕ)〉with
divS2 = −〈τ2(ϕ), dϕ〉(Jiang, Loubeau–M–O)
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The meaning of S2 = 0 (Loubeau–M–O)
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The meaning of S2 = 0 (Loubeau–M–O)
A smooth map ϕ : (M,g) → (N,h) is biharmonic if it is a criticalpoints of the bienergy w.r.t. variations of the map.
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The meaning of S2 = 0 (Loubeau–M–O)
A smooth map ϕ : (M,g) → (N,h) is biharmonic if it is a criticalpoints of the bienergy w.r.t. variations of the map.Opting for a different angle of attack, one can vary the metricinstead of the map and consider the functional
F : G → R, F (g) = E2(ϕ),
where G is the set of Riemannian metrics on M
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The meaning of S2 = 0 (Loubeau–M–O)
A smooth map ϕ : (M,g) → (N,h) is biharmonic if it is a criticalpoints of the bienergy w.r.t. variations of the map.Opting for a different angle of attack, one can vary the metricinstead of the map and consider the functional
F : G → R, F (g) = E2(ϕ),
where G is the set of Riemannian metrics on M
Theorem
δ(F (gt)) = −1
2
∫
M
〈S2, ω〉 vg,
The tensor S2 vanishes precisely at critical points of the energy(bienergy) for variations of the domain metric, rather than forvariations of the map.
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The condition S2 = 0 is rather strong, in fact
S2 = 0 ⇒ harmonic if:
• dim(M) = 2
• M is compact and orientable with dim(M) 6= 4
• ϕ is an isometric immersion and dim(M) 6= 4
• M is complete and ϕ has finite energy and bienergy
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Isometric immersion
If ϕ : (M,g) → (N,h) is an isometric immersion from
divS2 = −〈τ2(ϕ), dϕ〉⇓
div S2 = − τ2(ϕ)⊤
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Isometric immersion
If ϕ : (M,g) → (N,h) is an isometric immersion from
divS2 = −〈τ2(ϕ), dϕ〉⇓
div S2 = − τ2(ϕ)⊤
Problem
Study isometric immersions in space forms with divS2 = 0
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Biharmonic submanifolds in a Riemannian manifold
An isometric immersion
ϕ : (M,g) → (N,h)
is biharmonic iff
∆⊥H+ traceB(·, AH·) + trace(RN (·,H)·)⊥ = 0
m2grad |H|2 + 2 traceA∇⊥
(·)H(·) + 2 trace(RN (·,H)·)⊤ = 0
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Results for Bih. Sub. in non constant sec. curv.manifolds
• In three-dimensional homogeneous spaces (Thurston’s ge-ometries)
(Inoguchi, Ou–Wang, Caddeo–Piu–M–O)
• There exists examples of proper biharmonic hypersurfacesin a space with negative non constant sectional curvature
(Ou–Tang)
• It is initiated the study of biharmonic submanifolds in complexspace forms
(Ichiyama–Inoguchi–Urakawa, Fetcu–Loubeau–M–O, Sasahara)
• There are several works on biharmonic submanifolds in con-tact manifold and Sasakian space forms
(Inoguchi, Fetcu–O, Sasahara)
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In a Sasakian manifold
(N,Φ, ξ, η, g)
a submanifold M ⊂ N tangent to ξ is called anti-invariant if Φmaps any tangent vector to M , which is normal to ξ, to a vectorwhich is normal to M .
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Finite k-type submanifolds
An isometric immersion φ : M → Rn+1 (M compact) is called of
finite k-type ifφ = φ0 + φ1 + · · ·+ φk
where∆φi = λiφi, i = 1, . . . , k
and φ0 ∈ Rn+1 is the center of mass
A submanifold M ⊂ Sn ⊂ R
n+1 is said to be of finite type if it isof finite type as a submanifold of Rn+1.
A non null finite type submanifold in Sn is said to be mass-
symmetric if the constant vector φ0 of its spectral decompositionis the center of the hypersphere S
n, i.e. φ0 = 0.
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