Energy and volume of vector fields on spherical domains

0.09 MB·English

by  Fabiano G. B. Brito

ENERGY AND VOLUME OF VECTOR FIELDS ON SPHERICAL
DOMAINS
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1 FABIANOG.B.BRITO,ANDRE´ GOMES,ANDGIOVANNIS.NUNES
0
2
Abstract. Wepresentinthispapera“boundaryversion”fortheoremsabout
n minimality of volume and energy functionals on a spherical domain of three-
a dimensionalEuclideansphere.
J

7
2

1. Introduction
]
G
Let (M,g) be a closed, n-dimensional Riemannian manifold and T1M the unit
D
tangent bundle of M considered as a closed Riemannian manifold with the Sasaki
. metric. Let X : M T1M be a unit vector field defined on M, regarded as a
h
−→
t smooth section on the unit tangent bundle T1M. The volume of X was defined in
a
[8] by vol(X) := vol(X(M)), where vol(X(M)) is the volume of the submanifold
m
X(M) T1M. Using an orthonormal local frame e1,e2,...,en−1,en =X , the
[ ⊂ { }
volume of the unit vector field X is given by

1 n
v vol(X)= (1+ X 2+ X X 2+...
9 ZM Xa=1k∇ea k Xa<bk∇ea ∧∇eb k
5
2
2 ...+ X X )1/2ν (g)
1.5 a1<·X··<an−1(cid:13)(cid:13)(cid:13)∇ea1 ∧···∧∇ean−1 (cid:13)(cid:13)(cid:13) M
and the energy of the vector field X is given by
0
1 n 1 n
2
1 E(X)= 2vol(M)+ 2Z k∇eaXk νM(g)
: MXa=1
v
i The Hopf vector fields on S3 are unit vector fields tangent to the classical Hopf
X fibration π :S3 S2 with fiber homeomorphic to S1.
r The following th−e→orems gives a characterization of Hopf flows as absolute minima
a
of volume and energy functionals:
Theorem 1.1 ([8]). The unit vector fields of minimum volume on the sphere S3
are precisely the Hopf vector fields and no others.
Theorem 1.2 ([1]). The unit vector fields of minimum energy on the sphere S3
are precisely the Hopf vector fields and no others.
We prove in this paper the following boundary version for these Theorems:
Theorem 1.3. Let U be an open set of the three-dimensional unit sphere S3 and
let K U be a compact set. Let~v be an unit vector field on U which coincides with
⊂
a Hopf flow H along the boundary of K. Then vol(~v) vol(H) and (~v) (H).
≥ E ≥E
Key words and phrases. Differentialgeometry, energyandvolumeofvector fields.
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2 FABIANOG.B.BRITO,ANDRE´ GOMES,ANDGIOVANNIS.NUNES
Other results for higher dimensions may be found in [2], [5], [7] and [8].
2. Preliminaries
Let U S3 be an open set. We consider a compact set K U. Let H be a
Hopf vect⊂or field on S3 and let ~v be an unit vector field define⊂d on U. We also
consider the map ϕ~v : U S3(√1+t2) given by ϕ~v(x) = x+t~v(x). This map
t −→ t
was introduced in [10] and [3].
Lemma 2.1. For t>0 sufficiently small, the map ϕ~v is a diffeomorphism.
t
Proof. A simple application of the identity perturbation method (cid:3)
In order to find the Jacobian matrix of ϕu~, we define the unit vector field ~u
t
1 t
~u(x):= ~v(x) x
√1+t2 − √1+t2
Using an adapted orthonormal frame e ,e ,~v on a neighborhood V U, we ob-
1 2
{ } ⊂
tain an adapted orthonormal frame on ϕ~v(V) given by e¯ ,e¯ ,~u , where e¯ = e ,
t { 1 2 } 1 1
e¯ =e .
2 2
In this manner, we can write
dϕ~v(e ) = dϕ~v(e ),e e + dϕ~v(e ),e e + dϕ~v(e ),~u ~u
t 1 t 1 1 1 t 1 2 2 t 1
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
dϕ~v(e ) = dϕ~v(e ),e e + dϕ~v(e ),e e + dϕ~v(e ),~u ~u
t 2 t 2 1 1 t 2 2 2 t 2
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
dϕ~v(~v) = dϕ~v(~v),e e + dϕ~v(~v),e e + dϕ~v(~v),~u ~u
t t 1 1 t 2 2 t
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
Now, by Gauss’ equation of immersion S3 ֒ R4, we have
→
d~v(Y)= ~v ~v,Y x
Y
∇ −h i
for every vector field Y on S3, and then
dϕ~v(e ),e = e +td~v(e ),e =1+t ~v,e
t 1 1 h 1 1 1i h∇e1 1i
(cid:10) (cid:11)
Analogously, we can conclude that
dϕ~v(e ),e = t ~v,e
t 1 2 h∇e1 2i
(cid:10) (cid:11)
dϕ~v(e ),e = t ~v,e
t 2 1 h∇e2 1i
(cid:10) (cid:11)
dϕ~v(e ),e = 1+t ~v,e
t 2 2 h∇e2 2i
(cid:10) (cid:11)
dϕ~v(e ),~u = 0
t 1
(cid:10) (cid:11)
dϕ~v(e ),~u = 0
t 2
(cid:10) (cid:11)
dϕ~v(~v),~u = 1+t2
t
(cid:10) (cid:11) p
By applying the notation h (~v) := ~v,e (i,j = 1,2), the determinant of the
ij h∇ei ji
Jacobian matrix of ϕ~v can be express in the form
t
det(dϕ~v)= 1+t2(1+σ (~v).t+σ (~v).t2)
t 1 2
p
where, by definition,
σ (~v) := h (~v)+h (~v)
1 11 22
σ (~v) := h (~v)h (~v) h (~v)h (~v)
2 11 22 12 21
−
ENERGY AND VOLUME OF VECTOR FIELDS ON SPHERICAL DOMAINS 3
3. Proof of Theorem 1.3
The energy of the vector field~v (on K) is given by
1 3 1
2 2
(~v):= d~v = vol(K)+ ~v
E 2Z k k 2 2Z k∇ k
K K
Using the notations above, we have
2
3 1
(~v) = vol(K)+ [( (h )2+( ~v,e )2+( ~v,e )2]
E 2 2Z ij h∇~v 1i h∇~v 2i
K iX,j=1
and then
2
3 1
(~v) vol(K)+ (h )2
E ≥ 2 2Z ij
KiX,j=1
3 1
vol(K)+ 2(h h h h )
≥ 2 2Z 11 22− 12 21
K
3
= vol(K)+ σ (~v)
2 Z 2
K
On the other hand, by change of variables theorem, we obtain
vol[ϕH(K)]= 1+t2(1+σ (H).t+σ (H).t2)=δ vol(S3( 1+t2))
t ZKp 1 2 · p
where δ :=vol(K)/vol(S3).
(Remark that σ (H) and σ (H) are constant functions on S3, in fact, we have
1 2
σ (H)=0 and σ (H)=1, by a straightforwardcomputation shown in [6]).
1 2
Suppose now that ~v is an unit vector field on K which coincides with a Hopf
vector field H on the boundary of K. Then, obviously
vol[ϕ~v(K)]=vol[ϕH(K)]
t t
Therefore, we obtain
vol[ϕ~v(K)] = 1+t2(1+σ (~v).t+σ (~v).t2)
t Z 1 2
Kp
= δ vol(S3( 1+t2))=[vol(K)](1+t2)3/2
· p
By identity of polynomials, we conclude that
σ (~v)=vol(K)
Z 2
K
and consequently
3
(~v) vol(K)+vol(K)= (H)
E ≥ 2 E
Now, observing that
2
vol(H)=2vol(K), σ (~v)=vol(K) and h2(~v) 2σ (~v)
Z 2 ij ≥ 2
K iX,j=1
4 FABIANOG.B.BRITO,ANDRE´ GOMES,ANDGIOVANNIS.NUNES
we can obtain an analogue of this result for volumes
vol(X) = 1+ h2 +[det(h )]2+
ZKq X ij ij ···
1+2σ +σ2
≥ Z q 2 2
K
= (1+σ )=2vol(K)=vol(H) (cid:3)
Z 2
K
4. Final remarks
(1) If K is a spherical cap (the closure of a connected open set with round
boundaryofthe three unitsphere),the theoremprovidesa“boundaryver-
sion” for the minimalization theorem of energy and volume functionals on
[1] and [8].
(2) The “Hopf boundary” hypothesis is essential. In fact, if there is no con-
straint for the unit vector field ~v on ∂K, it is possible to construct vector
fields on “small caps” such that ~v is small on K (exponential maps
k∇ k
may be used on that construction). A consequence of this is that (~v) and
E
vol(~v) are less than volume and energy of Hopf vector fields respectively.
(3) The results of this paper may, possibly, be extended for the energy of
solenoidal unit vector fields in the higher dimensional case (n = 2k+1).
We intend to treat this subject in a forthcoming paper.
(4) We express our gratitude to Prof. Jaime Ripoll for helpful conversation
concerning the final draft of our paper.
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ENERGY AND VOLUME OF VECTOR FIELDS ON SPHERICAL DOMAINS 5
Universidade Federal doEstado do Rio de Janeiro,Brazil
Universidade de Sa˜oPaulo,Brazil
Universidade Federal de Pelotas, Brazil