Harmonic Vector Fields -...

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

• (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


(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


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


(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.


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)].


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 .


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.


• 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)) .


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 ∗.


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.


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.


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 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,


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?


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.


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 .


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.


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


∆ : 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.


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 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 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


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.


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).


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.


• 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.


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.


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.


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〉

).


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.


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.


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?


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?


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 ∇


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.


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

.


• 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.


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.


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).


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 .


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 .


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 ).


• 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.


Book presentation

Sorin DragomirDomenico Perrone

Variational Principles and Differential Geometry


Sorin Dragomir • Domenico Perrone


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


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