Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Harmonic Vector Fields
Sorin Dragomir
PRIN Workshop - PisaFebruary 28 - March 3, 2013
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Harmonic maps
• (M , g), (N , h) oriented Riemannian manifolds,
φ : M → N a C∞ map,
‖dφ‖2 = traceg (φ∗g) : M → [0,+∞)Hilbert-Schmidt norm of dφ,
EΩ(φ) =1
2
∫Ω
‖dφ‖2dvg , φ ∈ C∞(M ,N),
Dirichlet energy functional,
Ω ⊂⊂ M relatively compact domain,
dvg canonical volume form of (M , g) i.e. locally
dvg =√
G dx1 ∧ · · · ∧ dxn on U
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
(U , x i) local coordinate system on M ,
G = det[gij ], gij = g
(∂
∂x i,∂
∂x j
).
• φ ∈ C∞(M ,N) is a harmonic map ifd
dtEΩ(φt)t=0 = 0
∀ Ω ⊂⊂ M , ∀ φt|t|<ε ⊂ C∞(M ,N): φ0 = φ,
Supp(V ) ⊂ Ω, V =
(∂φt
∂t
)t=0
∈ C∞(M , φ−1T (N)),
φ−1T (N)→ M pullback bundle.
• From now on M compact and E = EM .
First variation formula:
d
dtE (φt)t=0 = −
∫M
hφ(τ(φ) , V )dvg
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hφ = φ−1h pullback of h,
τ(φ) = tracegβ(φ) ∈ C∞(M , φ−1T (N)) tension field,
β(φ) = ∇φXφ∗Y − φ∗∇XY second fundamental form of φ,
X ,Y ∈ C∞(M ,T (M)),
∇φ = φ−1∇N ∈ C(φ−1T (N)) pullback of ∇N ,
∇, ∇N Levi–Civita connections of (M , g), (N , h).
Hence
φ ∈ C∞(M ,N) is harmonic ⇐⇒ τ(φ) = 0.
τ(φ) = 0 Euler-Lagrange equations. Locally
τ(φ)α = ∆φα + g ij(Γαβγ φ
) ∂φβ∂x i
∂φγ
∂x j
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
(U , x i) local coordinate system on M ,[g ij]
= [gij ]−1,
(V , yα) local coordinate system on N , φα = yα φ,
Γαβγ Christoffel symbols of hαβ = h
(∂
∂yα,∂
∂yβ
),
∆u = −div (∇u) Laplace-Beltrami operator of (M , g),u ∈ C 2(M). Locally
∆u = − 1√G
∂
∂x i
(√G g ij ∂u
∂x j
)on U . Hence
u ∈ C∞(M ,N) is harmonic ⇐⇒ ∀ (U , x i), (V , yα):
∆φα + g ij(Γαβγ φ
) ∂φβ∂x i
∂φγ
∂x j= 0, 1 ≤ α ≤ ν, (1)
ν = dim(N), the harmonic map system.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Suffices φ ∈ C 2(M ,N) for all constructions; but then
(1) quasi-linear elliptic & φ harmonic =⇒ φ ∈ C∞(M ,N).
• Examples of harmonic maps:
- geodesics in Riemannian manifolds;
- minimal isometric immersions (among Riemannianmanifolds);
- Riemannian submersions with minimal fibres;
- harmonic morphisms i.e. φ : M → N continuous map:∀ v ∈ L1
loc(V ), V ⊂ N open, ∆Nv = 0 in V =⇒=⇒ ∆(v φ) = 0 in U = φ−1(V ).
[∀ p ∈ V , ∃ (V , yα) on N such that ∆Nyα = 0(local harmonic coordinates)
φ harmonic morphism =⇒ 0 = ∆(yα φ) = ∆φα =⇒=⇒ φα ∈ C∞(U) =⇒ φ ∈ C∞(M ,N)].
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Theorem (B. Fuglede & T. Ishihara, 1979)
If dim(M) = n ≥ ν then any harmonic morphism is aharmonic map.If n < ν harmonic morphisms are constant maps.
Proof based on:
Lemma
∀ p ∈ N and ∀ (V , yα) local system of normal coordinateswith p ∈ V and yα(p) = 0, and ∀ C , Cα , Cαβ ∈ R withCαβ = Cβα, ∃ v : V → R such that ∆Nv = 0 in V and
v(p) = C ,∂v
∂yα(p) = Cα ,
∂2v
∂yα ∂yβ(p) = Cαβ .
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Proof of Lemma 2 by Fluglede: based on a version of theimplicit function theorem in infinite dimension;
Proof of Lemma 2 by Ishihara: a mess.
Proof still correct (read together with work by Lipman Bers).
P. Baird & J.C. Wood, Harmonic morphisms betweenRiemannian manifolds, London Math. Soc. Monographs, NewSeries, Vol. 29, Clarendon Press, Oxford, 2003.
L. Bers, Local behavior of solutions of general linear ellipticequations, Comm. Pure Appl. Math., 8 (1955), 473-496.
B. Fuglede, Harmonic morphisms between Riemannianmanifolds, Ann. Inst. Fourier (Grenoble), 28(1978), 107-144.
T. Ishihara, A mapping of Riemannian manifolds which
preserves harmonic functions, J. Math. Kyoto Univ., 19(1979),
215-229.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
• Weak harmonic maps
First approach:
whole N covered by one ψ = (y 1, · · · , y ν) : N → Rν
W 1,2(M ,N) = φ : M → N |φα ∈ W 1,2(M), 1 ≤ α ≤ ν[u ∈ L1
loc(M) has a weak gradient if ∃ Yu ∈ L1loc(M ,T (M))
such that ∫M
g(Yu , X )dvg = −∫M
u div(X )dvg ,
X ∈ C∞0 (M ,T (M)) .
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Yu uniquely determined up to a set of measure zero, denotedby Yu = ∇u
D(∇) = W 1,2(M) space of all u ∈ L2(M) having a weakgradient ∇u ∈ L2(M ,T (M));
∇ : D(∇) ⊂ L2(M)→ L2(M ,T (M)) (densely defined linearoperator of Hilbert spaces)
∇∗ : D(∇∗) ⊂ L2(M ,T (M))→ L2(M) adjoint of ∇ i.e.
D(∇∗) space of all X ∈ L2(M ,T (M)) such that∃ X ∗ ∈ L2(M) with∫
M
g(∇u , X )dvg =
∫M
u X ∗dvg , u ∈ D(∇),
and ∇∗X = X ∗.
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C∞0 (M ,T (M)) ⊂ D(∇∗) and ∇∗|C∞0 (M,T (M)) = −div.
∆ : D(∆) ⊂ L2(M)→ L2(M) Laplacian i.e.
D(∆) = u ∈ D(∇) : ∇u ∈ D(∇∗) and ∆ = ∇∗ ∇].
Back to (M , g) compact orientable Riemannian;
φ ∈ W 1,2(M ,N) is a weak harmonic map if∫U
g ∗(∇φα , ∇ϕ) + g ij(Γαβγ φ)
∂φβ
∂x i
∂φγ
∂x jϕ
dvg = 0
∀ ϕ ∈ C∞0 (U), ∀ (U , x i) on M .
Fundamental problem: existence and (partial) regularity ofweak harmonic maps.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Dirichlet problem for harmonic map systemS. Hildebrand & H. Kaul & K. Widman, 1977
Theorem
i) Existence: N complete, ∂N = ∅, ν ≥ 2, Ω ⊂⊂ M domain
Sect(N) ≤ κ2, p ∈ N, µ < min π
2κ, i(p)
,
i(p) injectivity radius of p, ϕ ∈ C (Ω,N) ∩W 1,2(Ω,N)with ϕ(Ω) ⊂ B(p, µ). =⇒ ∃ uniqueφ ∈ W 1,2(Ω,N) ∩ L∞(Ω,N):φ(Ω) ⊂ B(p, µ), φ− ϕ ∈ W 1,2
0 (Ω,N),φ minimizes EΩ among all such maps,φ is a weak harmonic map.
ii) Interior regularity: φ : M → N bounded weak harmonic,φ(M) ⊂ B(p, µ) =⇒ φ ∈ C (M ,N).
No discussion here of higher interior regularity or boundaryregularity.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
F. Helein, Regularite des applications faiblement harmoniquesentre une surface et une variete riemannienne, C. R. Acad. Sci.Paris Ser. I Math. (8)312 (1991), 591-596.
F. Helein, Harmonic maps, conservation laws and movingframes, Cambridge Tracts in Mathematics, 150, CambridgeUniversity Press, Cambridge, 2002.
S. Hildebrandt, Harmonic mappings of Riemannian manifolds.Harmonic mappings and minimal immersions, (Montecatini,1984), 1-117, Lecture Notes in Math., 1161, Springer, Berlin,1985.
S. Hildebrandt & K-O. Widman, On the Holder continuity ofweak solutions of quasilinear elliptic systems of second order,Ann. Scuola Norm. Sup. Pisa Cl. Sci., (1)4(1977), 145-178.
S. Hildebrandt & H. Kaul & K-O. Widman, An existence
theorem for harmonic mappings of Riemannian manifolds,
Acta Math., (1-2)138(1977), 1-16.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Harmonic vector fields
R. Moser, Unique solvability of the Dirichlet problem forweakly harmonic maps, Manuscripta Math., (3)105(2001),379-399.
R. Schoen & K. Uhlenbeck, Regularity of minimizing harmonic
maps into the sphere, Invent. Math., (1)78(1984), 89-100.
• (M , g) compact oriented Riemannian manifold,
V : M → T (M) tangent vector field,
G Sasaki metric on T (M)
∀ X ,Y ∈ X(T (M)) :
G (X ,Y ) = g Π(LX , LY ) + g Π(KX ,KY ),
g Π = Π−1g pullback metric on Π−1T (M)→ M ,Π : T (M)→ M projection,
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L : T (T (M))→ Π−1T (M), LvX = (v , (dvΠ)X ),
K : T (T (M))→ Π−1T (M), Kv = γ−1v Qv Dombrowski map
X ∈ Tv (T (M)), v ∈ T (M), Qv : Tv (T (M))→ Ker(dvΠ)projection,
Tv (T (M)) = Hv ⊕Ker(dvΠ),
H horizontal distribution on T (M) associated to theLevi–Civita connection ∇ of (M , g).
• Hence V smooth map of Riemannian manifolds (M , g) and(T (M),G ).
Harmonicity?
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Theorem (T. Ishihara & O. Nouhaud, 1977)
The following are equivalent
i) V : (M , g)→ (T (M),G ) is a harmonic map.
ii) V is an absolute minimum of
E (X ) =1
2
∫M
‖dX‖2dvg , X ∈ X(M).
iii) ∇V = 0.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
S. Dragomir & D. Perrone, Harmonic vector fields. Variationalprinciples and differential geometry, Elsevier Inc.,Amsterdam-Boston-Heidelberg-London-NewYork-Oxford-Paris-San Diego-SanFrancisco-Singapore-Sydney-Tokyo, 2012.
T. Ishihara, Harmonic sections of tangent bundles, J. Math.Tokushima Univ., 13(1979), 23-27.
O. Nouhaud, Applications harmoniques d’une variete
Riemannienne dans son fibre tangent, C.R. Acad. Sci. Paris,
284(1977), 815-818.
• Yet:
E (V ) =n
2Vol(M) + B(V ), B(V ) =
1
2
∫M
‖∇V ‖2dvg .
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Total bending functional, or biegung =⇒ parallel vector fieldsare trivially harmonic maps.
Thus
Domain of E : C∞(M ,T (M))→ [0,+∞) too large.
Look for critical points of E : X(M)→ [0,+∞)
[variations of V ∈ X(M) are through vector fieldsVt|t|<ε ⊂ X(M) with V0 = V ]
Theorem (O. Gil-Medrano, 2001)
V ∈ X(M) critical point of E : X(M)→ R =⇒ ∇V = 0.
O. Gil-Medrano, Relationship between volume and energy of
unit vector fields, Diff. Geometry Appl., 15(2001), 137-152.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Hence: Domain of E : X(M)→ [0,+∞) still too large.
• U(M , g)x = v ∈ Tx(M) : gx(v , v) = 1, x ∈ M ,
Sn−1 → U(M , g)→ M tangent sphere bundle
X1(M) = C∞(U(M , g)) unit vector fields
X ∈ X1(M) is a harmonic vector field if critical point ofE : X1(M)→ [0,+∞) [variations through unit vector fields]
• First variation formula
d
dtE (Xt)t=0 =
∫M
g(∆X ,V )dvg
Xt|t|<ε ⊂ X1(M), X0 = X ,
Vx =d
dtt 7→ Xtt=0 ∈ Tx(M), g(X ,V ) = 0
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∆ : X(M)→ X(M) rough Laplacian
∆X = −n∑
i=1
∇Ei∇Ei
X −∇∇EiEi
X
,
Ei : 1 ≤ i ≤ n local g -orthonormal frame of T (M).
Symbol of rough Laplacian σ2(∆)ω : Tx(M)→ Tx(M),
ω ∈ T ∗x (M) \ 0, x ∈ M , σ2(∆)ω(v) = ‖ω‖2 v , v ∈ Tx(M).
σ2(∆)ω isomorphism =⇒ ∆ elliptic.
• Euler–Lagrange equations of constrained variational principle
d
dtE (Xt)t=0 = 0, g(X ,X ) = 1,
are∆X − ‖∇X‖2X = 0.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
S. Dragomir & D. Perrone, On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields, Ann.Global Anal. Geom., 30(2006), 211-238.
G. Wiegmink, Total bending of vector fields on Riemannianmanifolds, Math. Ann., (2)203)(1995), 325-344.
G. Wiegmink, Total bending of vector fields on the sphere S3,Diff. Geom. Appl., 6(1996), 219-236.
C.M. Wood, On the energy of a unit vector field, Geom.Dedicata, 64(1997), 319-330.
C.M. Wood, The energy of Hopf vector fields, Manuscripta
Math., 101(2000), 71-78.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Harmonic vector fields X : M → U(M , g) aren’t harmonicmaps (M , g)→ (U(M , g),G ) in general:
Theorem (S.D. Han & J.W. Yim, 1998)
Tension field of X : (M , g)→ (U(M , g),G )
τ(X ) =tracegR (∇·X , X ) · H − tan (∆X )V
X .
Hence
X harmonic map ⇐⇒i) ∆X − ‖∇X‖2X = 0 and
ii) traceg R(∇·X , X )· = 0.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Harmonic vector fields on Riemannian tori
S.D. Han & J.W. Yim, Unit vector fields on spheres which are
harmonic maps, Math. Z., 227(1998), 83-92.
• d1, d2 ∈ R2 linearly independent
Γ = m d1 + n d2 ∈ R2 : m, n ∈ Z lattice
T 2 = R2/Γ torus
π : R2 → T 2 projection
Assume: T 2 oriented, π : R2 → T 2 orientation preserving
J almost complex structure on T 2 (induced by fixedorientation)
g Riemannian metric on T 2
S ,W ⊂ T (T 2) g -orthonormal frame with W = JS
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S1 → S(T 2, g)→ T 2 tangent sphere bundle
E = Γ∞(S(T 2, g)) • X ∈ E :
ϕ, ψ ∈ C∞(T 2,R) the (S ,W )-coordinates of X i.e.
ϕ = g(X , S), ψ = g(X ,W ),
X = ϕ S + ψW , ϕ2 + ψ2 = 1.
E → C∞(T 2, S1), X 7→ ϕ +√−1ψ bijection
• α : R2 → R is an (S ,W )-angle function for X if
X π = (cosα) S π + (sinα) W π.
∀ (m, n) ∈ Z2 set
Per(m, n) =α ∈ C∞(R2) : ∀ ξ ∈ R2
α(ξ + d1)− α(ξ) = 2mπ, α(ξ + d2)− α(ξ) = 2nπ
• α ∈ Per(m, n) is a (m, n)-semiperiodic function.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Also set
W =⋃
(m,n)∈Z2
Per(m, n).
Lemma
a) ∀ X ∈ E : ∃ an angle function α ∈ W .
X ∈ E =⇒ angle functions of X differ by integer multiples of2π and lie in but one Per(m, n) for some (m, n) ∈ Z2.
b) Let α ∈ C∞(R2). Then α is an angle function for someX ∈ E ⇐⇒ α ∈ W .
Define htp(S ,W ) : E → Z2 as follows:
Let X ∈ E Lemma 7 =⇒∃ unique (m, n) ∈ Z2: angle functions of X ⊂ Per(m, n);
Set htp(S,W )(X ) = (m, n).
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Lemma
X ,Y ∈ E homotopic in E ⇐⇒ htp(S ,W )(X ) = htp(S ,W )(Y ).Homotopy classes of elements of E are thus classified by theelements of Z2.
• X topological space
π1(X) homotopy classes of maps f : X→ S1 (theBruschlinsky group); X ∈ E : (m, n) = htp(S ,W )(X ) ∈ Z2
α ∈ Per(m, n) angle function for X
Let e iα : T 2 → S1,(e iα)
(p) = e i α(ξ) , ξ ∈ π−1(p), p ∈ T 2
=⇒ E → C∞(T 2, S1), X 7→ e iα bijection
=⇒ [X ] : X ∈ E ≈ π1(T 2) group isomorphism
Hence Lemma 8 is the calculation of the Bruschlinsky group ofthe torus i.e. π1(T 2) = Z⊕ Z.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
• biegung in terms of angle functions:
g(∇S , W ) = a g(S , ·) + b g(W , ·), a, b ∈ C∞(T 2),
S , W ∈ X(R2): π-related to S , W
Z = (a π) S + (b π) W ∈ X(R2)
Q = sd1 + td2 ∈ R2 : (s, t) ∈ [0, 1]2g = π∗g
B(X ) =
∫T 2
‖∇X‖2dvg =
∫Q
‖∇α + Z‖2g π∗dvg .
May think of B :W → [0,+∞).
∃ Action of R on W : R×W →W , (r , α) 7→ α + r .
Action leaves B invariant and preserves Per(m, n) ⊂ W ,∀ (m, n) ∈ Z2.
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Read on E = C∞(S(T 2, g)) action is SO(2)× E → E((cos r − sin rsin r cos r
), X
)7→ (cos r) X + (sin r) JX .
Theorem (G. Wiegmink, 1995)
a) The following statements are equivalent
i) X ∈ Crit(B).ii) ∀ angle function α of X
∆α− (Sa + Wb) π = 0. (2)
iii) (S ,W )-Coordinates (ϕ,ψ) of X satisfy
ϕ∆ψ − ψ∆ϕ− Sa−Wb = 0.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Theorem
b) If X ∈ Crit(B) then the full orbit of X under any smoothaction of a Lie group on E leaving B invariant consists ofcritical points. The set of critical points of B intersectseach homotopy class E (S,W )
(m,n) ∈ [X ] : X ∈ E ⊂π(T 2, S(T 2)) exactly in one orbit of the SO(2)-action onE . Therefore, up to this action, there is but one criticalpoint in each class E (S ,W )
(m,n) .
c) Let u ∈ C∞(T 2) and let g = e2ug be a metric on T 2 inthe conformal class of g . Then a unit vector field X ∈ Eis a critical point of B if and only if e−uX is a criticalpoint of B.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Theorem
d) Let h be a flat metric on T 2 and ∇h its Levi–Civitaconnection. There is a h-orthonormal frame S0,W0which is parallel with respect to ∇h. Let 〈·, ·〉 and ‖ · ‖be the Euclidean inner product and norm on E2; let usset D =‖d1‖2‖d2‖2−〈d1, d2〉2. The (S0,W0)-anglefunctions λm,n : R2 → R of the critical points of B on
(T 2, h) in the homotopy class E (S0,W0)(m,n) are given by
λm,n(ξ)=2π
D
[〈d1, ξ〉
(m‖d2‖2 − n〈d1, d2〉
)+ (3)
+〈d2, ξ〉(n‖d1‖2 −m〈d1, d2〉
)]+ s, s ∈ R,
for any ξ ∈ R2. Also
B(λm,n) =π
D
(m2‖d2‖2 + n2‖d1‖2 − 2mn〈d1, d2〉
).
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Theorem
e) For any critical point X ∈ E of B on (T 2, g) and for anyq ∈ T 2 there is a conformal coordinate chart f : U → E2
(where U ⊆ T 2 is an open neighborhood of q) such that
X = (cosλm,n)∂
∂f1+ (sinλm,n)
∂
∂f2
in terms of the Gaussian frame field of f with λm,n as in(3) for suitable (m, n) ∈ Z2. When (m, n) = (0, 0) thef -coordinates of X are constant.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Stability
• (M , g) compact orientable Riemannian manifold
X ∈ X1(M) harmonic vector field
X : M × I 2δ → S(M , g), Iδ = (−δ, δ), δ > 0,
Xt,s(x) = X(x , t, s), x ∈ M , t, s ∈ Iδ, X0,0 = X ,
V =
(∂Xt,s
∂t
)t=s=0
, W =
(∂Xt,s
∂s
)t=s=0
∂2
∂t ∂sB(Xt,s)t=s=0 =
∫M
g(V , ∆W − ‖∇X‖2W )dvg
second variation formula
Hence: a stability theory for harmonic vector fields.
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Theorem (G. Wiegmink, 1995)
(T 2, g) Riemannian torusX0 ∈ Crit(B) ⊂ E =⇒ ∀ smooth 1-parameter variationXt|t|<ε ⊂ E of X
d2
dt2B(X (t))t=0 ≥ 0. (4)
Equality in (4) ⇐⇒ Xt|t|<ε is in first order contact at t = 0with a variation Yt|t|<ε ⊂ E of X0 such thatY (t) ∈ SO(2) · X0 for any |t| < ε.
Question: Stability of harmonic vector fields related tospectrum of rough Laplacian?
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Analogs of G. Wiegmink’s results for harmonic vector fields onLorentzian surfaces are available:
S. Dragomir & M. Soret, Harmonic vector fields on compact
Lorentz surfaces, Ricerche mat. DOI
10.1007/s11587-011-0113-1
Problem: Study biharmonic vector fields i.e. unit vector fieldsX which are critical points of the bi-energy functional
E2(X ) =1
2
∫Ω
‖τ(X )‖2 d vg
X ∈ X1(M), Ω ⊂⊂ M ,
(variations through unit vector fields).When M = T 2 is a Riemannian torus, does each homotopyclass E (S ,W )
(m,n) contain a biharmonic representative?
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Weakly harmonic vector fields
Weak covariant derivatives
• (M , g) Riemannian manifold
T r ,s(M)→ M vector bundle of tangent (r , s)-tensor fields
∀ ϕ ∈ Γ(T r ,s(M)):
‖ϕ‖ = g ∗(ϕ, ϕ)1/2 : M → [0,+∞) is well defined.
We set
(ϕ, ψ) =
∫M
g ∗(ϕ, ψ)dvg
∀ ϕ, ψ ∈ Γ(T r ,s(M)) with g ∗(ϕ, ψ) ∈ L1(M).
∇ : C∞(T r ,s(M))→ C∞(T r ,s+1(M)) covariant derivative
∇∗ : C∞0 (T r ,s+1(M))→ C∞0 (T r ,s(M)) formal adjoint of ∇
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
i.e.(∇∗h , ϕ) = (h , ∇ϕ),
h ∈ C∞0 (T r ,s+1(M)), ϕ ∈ C∞0 (T r ,s(M)).
∀ p ≥ 1, Lp(T r ,s(M)): (r , s)-tensor fields ϕ with
‖ϕ‖Lp(T r,s(M)) =
(∫M
‖ϕ‖pdvg
)1/p
<∞.
ϕ ∈ L1loc(T r ,s(M)) is weakly differentiable
if ∃ ψ ∈ L1loc(T r ,s+1(M)):
(ψ , h) = (ϕ , ∇∗h), h ∈ C∞0 (T r ,s+1(M)).
ψ uniquely determined up to a set of measure zero;
Notation: ψ = ∇ϕ weak covariant derivative of ϕ.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Weakly harmonic vector fields
Sobolev spaces of tensor fields
• H0,pg (T r ,s(M)) = Lp(T r ,s(M))
H1,pg (T r ,s−1(M)) = ϕ ∈ H0,p
g (T r ,s−1(M)) :
ϕ weakly differentiable and ∇ϕ ∈ Lp(T r ,s(M)).Recursively ∀ k ≥ 2:
Hk,pg (T r ,s−1(M)) = ϕ ∈ Hk−1,p
g (T r ,s−1(M)) :
∇ϕ ∈ Hk−1,pg (T r ,s(M)).
Hk,pg (T r ,s(M)) Banach space with
‖ϕ‖Hk,pg (T r,s(M)) =
(k∑
j=0
‖∇jϕ‖pLp(T 0,j (M)⊗T r,s(M))
)1/p
.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
• In particular
H1,pg (T (M))=X ∈ Lp(T (M)) : ∇X ∈ Lp(T ∗(M)⊗T (M)),
‖X‖H1,pg (T (M)) =
(‖X‖pLp(T (M)) + ‖∇X‖pLp(T∗(M)⊗T (M))
)1/p
.
Theorem
1 ≤ p <∞ =⇒ H1,pg (T (M)) separable Banach;
1 < p <∞ =⇒ reflexive;
p = 2 =⇒ H1,2g (T (M)) separable Hilbert space.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
X ∈ H1,2g (T (M)) is a weak solution to ∆X − ‖∇X‖2X = 0 if∫
M
g ∗(∇X , ∇Y )− ‖∇X‖2 g(X ,Y )
dvg = 0
∀ Y ∈ X∞0 (M).
X ∈ H1,2g (T (M)) unit vector field
X weakly harmonic vector field if
weak solution to harmonic vector fields system.
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Examples
Radial vector fields
p ∈ M , U ⊂ M normal coordinate neighborhood of p
=⇒ r = dist(p , ·) : U \ p → R is smooth.
∂
∂r∈ X(U \ p) radial vector field i.e.
g
(∂
∂r, X
)= X (r), ∀ X ∈ X(U \ p).
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Radial vector field∂
∂ris
- unit vector field tangent to geodesics issuing at p;
- outward normal of small geodesic sphere S(p, a).
[Pfaffian system
(R∂
∂r
)⊥completely integrable and
geodesic spheres S(p, a) maximal integral manifolds]
- weakly harmonic vector field on U .
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
Normal vector fields on principal orbits
Theorem (G. Nunes & J. Ripoll, 2008)
(M , g) compact orientable Riemannnian manifold, n ≥ 3,G ⊂ Isom(M , g) compact Lie group acting on M withcohomogeneity one. Assume either G has no singular orbits oreach singular orbit of G has dimension ≤ n − 3.N unit vector field orthogonal to principal orbits of G =⇒N ∈ H1,2
g (T (M)) and critical point of biegung
B : H1,2g (T (M))→ R.
M∗ ⊂ M union of principal orbits of G ; H : M∗ → R, H(x) =mean curvature of orbit G (x) with respect to N=⇒ H ∈ L2(M) and total bending of N is
B(N) = −∫M
Ric(N ,N)dvg +
∫M
H2dvg .
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
QuestionExistence and partial regularity theory
for weakly harmonic vector fields?
Further open problems:• Given a Riemannian manifold (M , g) and X ∈ X(M) andω ∈ Ω1(M), let X [ ∈ Ω1(M) and ω] ∈ X(M) be given by
g(X ,Y ) = X [(Y ), g(ω],Y ) = ω(Y ), ∀ Y ∈ X(M).
Study smooth maps φ : M → N of Riemannian manifolds suchthat for any harmonic vector field Y ∈ X1(V ), defined on the
open set V ⊂ N , the vector field(φ∗Y [
)]is harmonic on
U = φ−1(V ).
Harmonic maps Harmonic vector fields Harmonic vector fields on tori Stability Weakly harmonic vector fields Examples Book
• Solve the Dirichlet problem for the harmonic vector fieldssystem on a domain Ω ⊂ M . See also
E. Barletta, On the Dirichlet problem for the harmonic vector
fields equation, Nonlinear Analysis, 67(2007), 1831-1846.
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Book presentation
Sorin DragomirDomenico Perrone
Variational Principles and Differential Geometry
Harmonic Vector Fields
Sorin Dragomir • Domenico Perrone
Harmonic Vector Fields
DraGomirPerrone
An essential tool for researchers in differential geometry, Harmonic Vector Fields: Variational Principles and Differential Geometry is devoted to the theory of harmonic vector fields on Riemannian, contact, CR, and Lorentzian manifolds. Although it is focused on the differential geometric properties of harmonic vector fields, this unique book carefully reports on interdisciplinary aspects, relating the subject to both nonlinear analysis (weak solutions to the harmonic vector fields equation) and analysis in several complex variables (subelliptic harmonic vector fields and tangential Cauchy-Riemann equations).
Key Features• Useful for any scientist familiar with the theory of harmonic maps
• A clear and rigorous exposition of the main results in the theory of harmonic vector fields, both old and new
• Provides applications to other mathematical disciplines, such as nonlinear partial differential equations, variational calculus, complex analysis in several complex variables, and general relativity
Variational Principles and Differential Geometry
Harmonic Vector Fields
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