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Axioms 2018, 7(4), 88; doi:10.3390/axioms7040088

Article
Exponentially Harmonic Maps into Spheres
1
Dipartimento di Matematica, Informatica, ed Economia, Università degli Studi della Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy
2
Dipartimento di Matematica e Fisica Ennio De Giorgi, Università del Salento, 73100 Lecce, Italy
*
Author to whom correspondence should be addressed.
Received: 29 October 2018 / Accepted: 18 November 2018 / Published: 22 November 2018

Abstract

:
We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere S m R m + 1 . Given a codimension two totally geodesic submanifold Σ S m , we show that every nonconstant exponentially harmonic map ϕ : M S m either meets or links Σ . If H 1 ( M , Z ) = 0 then ϕ ( M ) Σ .
Keywords:
exponentially harmonic map; totally geodesic submanifold; Euler-Lagrange equations

1. Introduction

Let M be a compact, connected, orientable n-dimensional Riemannian manifold, with the Riemannian metric g. Let ϕ : M N be a C map into another Riemannian manifold ( N , h ) . The Hilbert-Schmidt norm of d ϕ is d ϕ = trace g ϕ h 1 / 2 : M R . Let us consider the functional
E : C ( M , N ) R , E ( ϕ ) = M exp 1 2 d ϕ 2 d v g .
A C map ϕ : M N is exponentially harmonic if it is a critical point of E i.e., d E ( ϕ s ) / d s s = 0 = 0 for any smooth 1-parameter variation { ϕ s } | s | < ϵ C ( M , N ) of ϕ 0 = ϕ . Exponentially harmonic maps were first studied by J. Eells & L. Lemaire [1], who derived the first variation formula
d d s E ( ϕ s ) s = 0 = M exp e ( ϕ ) h ϕ V , τ ( ϕ ) + ϕ e ( ϕ ) d v g
where e ( ϕ ) = 1 2 d ϕ 2 and τ ( ϕ ) C ϕ 1 T N is the tension field of ϕ (cf. e.g., [2]). Also V = ϕ s / s s = 0 is the infinitesimal variation induced by the given 1-parameter variation. In particular, the Euler-Lagrange equations of the variational principle δ E ( ϕ ) = 0 are
Δ ϕ i + Γ j k i ϕ ϕ j x α ϕ k x β g α β + ϕ i x α e ( ϕ ) x β g α β = 0
where
Δ u = 1 G x α G g α β u x β , G = det [ g α β ] ,
is the Laplace-Beltrami operator and Γ j k i are the Christoffel symbols of h i j . The (partial) regularity of weak solutions to (1) was investigated by D.M. Duc & J. Eells (cf. [3]) when N = R and by Y-J. Chiang et al. (cf. [4]) when N = S m . Differential geometric properties of exponentially harmonic maps, including the second variation formula for E, were found by M-C. Hong (cf. [5]), J-Q. Hong & Y. Yang (cf. [6]), L-F. Cheung & P-F. Leung (cf. [7]), and Y-J. Chiang (cf. [8]).
The purpose of the present paper is to further study exponentially harmonic maps ϕ winding in N = S m , a situation previously attacked in [4], though confined to the case where M is a Fefferman space-time (cf. [9]) over the Heisenberg group H n and ϕ : M S m is S 1 invariant. Fefferman spaces are Lorentzian manifolds and exponentially harmonic maps of this sort are usually referred to as exponential wave maps (cf. e.g., Y-J. Chiang & Y-H. Yang, [10]). Base maps f : H n S m associated (by the S 1 invariance) to ϕ : M S m turn out to be solutions to degenerate elliptic equations [resembling (cf. [11]) the exponentially harmonic map system (1)] and the main result in [4] is got by applying regularity theory within subelliptic theory (cf. e.g., [12]).
Through this paper, M will be a compact Riemannian manifold and ϕ : M S m an exponentially harmonic map. Although the properties of an exponentially harmonic map may differ consistently from those of ordinary harmonic maps (see the emphasis by Y-J. Chiang, [13]), we succeed in recovering, to the setting of exponentially harmonic maps, the result by B. Solomon (cf. [14]) that for any nonconstant harmonic map ϕ : M S m from a compact Riemannian manifold either ϕ ( M ) Σ or ϕ : M S m Σ isn’t homotopically null. Here Σ S m is an arbitrary codimension 2 totally geodesic submanifold.
The ingredients in the proof of the exponentially harmonic analog to Solomon’s theorem (see [14]) are (i) setting the Equation (1) in divergence form
exp e ( ϕ ) ϕ i + 2 e ( ϕ ) exp e ( ϕ ) ϕ i = 0
(got by a verbatim repetition of arguments in [4]), (ii) observing that S m Σ is isometric to the warped product manifold S + m 1 × w S 1 , and (iii) applying the Hopf maximum principle (to conclude that there are no nonconstant exponentially harmonic maps into hemispheres).

2. Exponentially Harmonic Maps into Warped Products

Let S = L × R , where L is a Riemannian manifold with the Riemannian metric g L . Let w C ( S ) such that w ( y ) > 0 for any y S and let us endow S with the warped product metric
h = Π 1 g L + w 2 d t d t ,
where t = t ˜ Π 2 , t ˜ is the Cartesian coordinate on R , and
Π 1 : S L , Π 2 : S R ,
are projections. The Riemannian manifold ( S , h ) is customarily denoted by L × w R . Let ϕ : M S be an exponentially harmonic map and let us set
F = Π 1 ϕ , u = Π 2 ϕ .
We need to establish the following
Lemma 1.
Let M be a compact, connected, orientable Riemannian manifold and ϕ = ( F , u ) : M S = L × w R a nonconstant exponentially harmonic map. Then u is a solution to
w ϕ Δ u + w t ϕ u 2
= w ϕ ( u ) e ( ϕ ) + 2 ( u ) ( w ϕ ) .
If additionally w / t = 0 then ϕ ( M ) L × { t ϕ } for some t ϕ R .
Also for an arbitrary test function φ C ( M ) we set
ϕ s ( x ) = F ( x ) , u ( x ) + s φ ( x ) , x M , | s | < ϵ ,
so that { ϕ s } | s | < ϵ is a 1-parameter variation of ϕ . For each x 0 M let { E α : 1 α n } C ( U , T ( M ) ) be a local g-orthonormal (i.e., g ( E α , E β ) = δ α β ) frame, defined on an open neighborhood U M of x 0 . Then
d ϕ s 2 = trace g ϕ s h = α = 1 n ϕ s h ( E α , E α )
on U. On the other hand
ϕ s h ( X , X ) = F g L ( X , X ) + ( w ϕ s ) 2 X ( u ) + s X ( φ ) 2
for every tangent vector field X X ( M ) . Formula (3) for X = E α yields
d ϕ s 2 = d F 2 + w ϕ s 2 u 2 + 2 s g ( u , φ ) + s 2 φ 2 .
Hence (differentiating with respect to s)
d d s E ( ϕ s ) s = 0 = M exp e ( ϕ ) w ϕ 2 g ( u , φ )
+ w ϕ w t ϕ φ u 2 d v g
where w t = w / t . Moreover
exp e ( ϕ ) w ϕ 2 g u , φ
= div φ exp e ( ϕ ) ( w ϕ ) 2 u
+ φ exp e ( ϕ ) w ϕ 2 Δ u ( u ) exp e ( ϕ ) ( w ϕ ) 2
where div : X ( M ) C ( M ) is the divergence operator with respect to the Riemannian volume form
d v g = G d x 1 d x n
i.e., L X d v g = div ( X ) d v g and Δ is the Laplace-Beltrami operator (on functions) i.e., Δ u = div ( u ) . Substitution from (5) into (4) together with Green’s lemma yields [by { d E ( ϕ s ) / d s } s = 0 = 0 and the density of C ( M ) in L 2 ( M ) ]
w ϕ Δ u + w t ϕ u 2
= w ϕ ( u ) e ( ϕ ) + 2 ( u ) ( w ϕ )
which is (2) in Lemma 1. When w t = 0 Equation (6) is
div exp e ( ϕ ) ( w ϕ ) 2 u = 0 .
Equation (7) is part of the Euler-Lagrange system associated to the variational principle δ E ( ϕ ) = 0 . Next (by (7))
div w ϕ 2 u exp e ( ϕ ) u = exp e ( ϕ ) w ϕ 2 u 2 .
Let us integrate over M in (8) and use Green’s lemma. We obtain
M exp e ( ϕ ) w ϕ 2 u 2 d v g = 0
yielding (as ϕ is assumed to be nonconstant) u ( x ) = t ϕ for some t ϕ R and any x M . Q.e.d.

3. Exponentially Harmonic Maps Omitting a Codimension 2 Sphere Aren’t Null Homotopic

Let Σ S m be a codimension 2 totally geodesic submanifold. A continuous map ϕ : M S m meets Σ if ϕ ( M ) Σ and links Σ if ϕ ( M ) Σ = and ϕ : M S m Σ is not null homotopic. The purpose of the section is to establish
Theorem 1.
Let ϕ : M S m be a nonconstant exponentially harmonic map from a compact, connected, orientable Riemannian manifold M into the sphere S m R m + 1 . If Σ S m is a codimension 2 totally geodesic submanifold, then φ either meets or links Σ.
Proof. 
The proof is by contradiction, i.e., we assume that ϕ doesn’t meet Σ and the map ϕ : M S m Σ is null homotpic. Let ( ξ j ) be a system of coordinates on R m + 1 such that Σ is given by the equations ξ 1 = ξ 2 = 0 . Let S + m 1 R m be the hemisphere
S + m 1 = y = ( y , y m ) R m 1 × R : y S m 1 , y m > 0 .
Let us consider the map
I : S + m 1 × S 1 S m Σ , I ( y , ζ ) = y m u , y m v , y ,
y = y , y m S + m 1 , ζ = u + i v S 1 C .
Let g N denote the canonical Riemannian metric on S N R N + 1 . The map I is an isometry of S + m 1 × f S 1 onto ( S m Σ , g m ) with the warping function
f C ( S + m 1 × S 1 ) , f ( y , ζ ) = y m .
Let us consider the map ψ ˜ = I 1 ϕ . We need the following. □
Lemma 2.
Let S and S ¯ be Riemannian manifolds, π : S S ¯ a local isometry, and f ¯ : M S ¯ an exponentially harmonic map. Then every map f : M S such that π f = f ¯ is exponentially harmonic.
Proof. 
Let h and h ¯ be the Riemannian metrics on S and S ¯ . For every 1-parameter variation { f s } | s | < ϵ of f 0 = f we set f ¯ s = π f s so that { f ¯ s } | s | < ϵ is a 1-parameter variation of f ¯ 0 = f ¯ . A calculation relying on π h ¯ = h yields E ( f s ) = E ( f ¯ s ) for every | s | < ϵ . Q.e.d.
By Lemma 2 the map ψ ˜ = I 1 ϕ is exponentially harmonic. Let us set
F = π 1 ψ ˜ , u ˜ = π 2 ψ ˜ ,
where π 1 : S + m 1 × S 1 S + m 1 and π 2 : S + m 1 × S 1 S 1 are projections. Next let us consider a point x 0 M and set ζ 0 = u ˜ ( x 0 ) S 1 . Also, considered the covering map p : R S 1 , p ( t ) = exp ( 2 π i t ) , pick t 0 R such that p ( t 0 ) = ζ 0 . As ϕ is null homotopic, the map ψ ˜ is null homotopic as well. Then
u ˜ π 1 ( M , x 0 ) = 0
where π 1 ( M , x 0 ) is the first homotopy group of M. Consequently there is a unique smooth function u : M R such that p u = u ˜ and u ( x 0 ) = t 0 . The map
ψ = ( F , u ) : M S + m 1 × w R
is exponentially harmonic [because ψ = π ψ ˜ and
π = 1 S + m 1 , p : S + m 1 × w R S + m 1 × f S 1
is a local isometry, where w C ( S + m 1 ) is given by w ( y ) = y m ]. We may then apply Lemma 1 to the map ψ with L = S + m 1 to conclude that
ψ ( M ) S + m 1 × t ψ
for some t ψ R . It follows that F = π 1 ψ : M S + m 1 is exponentially harmonic. We shall close the proof of Theorem 1 by showing that exponentially harmonic mappings into S + m 1 are constant. □

4. Exponentially Harmonic Map System in Divergence Form

Let us consider the L 2 inner products
( u , v ) L 2 = M u v d v g , ( X , Y ) L 2 = M g ( X , Y ) d v g .
Let us think of the gradient ∇ as a first order differential operator : C 1 ( M ) C T ( M ) and let be its formal adjoint, i.e.,
X , u L 2 = X , u L 2
for any X C 1 T ( M ) and u C 1 ( M ) . Ordinary integration by parts shows that X = div ( X ) . Let ρ = exp e ( F ) C ( M ) . Starting from Δ u = div ( u ) one has
ρ Δ u , φ L 2 = u , ρ φ L 2 = u , ( ρ φ ) L 2
= ( ρ u ) , φ L 2 + M φ g ( u , ρ ) d v g
for any φ C ( M ) , that is
exp e ( F ) Δ u = exp e ( F ) u
+ exp e ( F ) g u , e ( F ) .
Lemma 3.
Let F : M S + m 1 be an exponentially harmonic map and F = j F where j : S m 1 R m is the inclusion. If F = F 1 , , F m then
exp e ( F ) F i + 2 e ( F ) exp e ( F ) F i = 0
for any 1 i m .
Proof. 
Let y = y 1 , , y m 1 : S + m 1 B m 1 be the projection, where B m 1 R m 1 is the open unit ball. With respect to this choice of local coordinates, the standard metric g m 1 and its Christoffel symbols are
h i j = δ i j + y i y j 1 | y | 2 , | y | 2 = i = 1 m 1 y i 2 ,
h i j = δ i j y i y j ,
Γ j k i = y i h j k .
Let us substitute from (13) into (1) [with ϕ i = F i ] and take into account
e ( F ) = 1 2 g α β F j x α F k x β h j k F .
The exponentially harmonic map system (1) becomes
Δ F i + 2 e ( F ) F i + g e ( F ) , F i = 0 , 1 i m 1 .
Multiplication of (15) by exp e ( F ) and subtraction from (9) [with u = F i ] yields (10) for any 1 i m 1 .
To see that (15) (and therefore (10)) holds for i = m as well, one first exploits the constraint ( F m ) 2 = 1 i = 1 m 1 ( F i ) 2 together with (11) and (14) to show that
e ( F ) = 1 2 j = 1 m F j 2 .
Finally, one contracts (15) by F i and uses once again the constraint together with Δ ( u 2 ) = 2 u Δ u u 2 . Q.e.d.
We may now end the proof of Theorem 1 as follows. Let F : M S + m 1 be an exponentially harmonic map. Let us integrate over M in (10) for j = m . Then (by Green’s lemma)
M e ( F ) exp e ( F ) F m d v g = 0
and F m > 0 so that
0 = e ( F ) = 1 2 j = 1 m F j 2
yielding F j = constant. So ϕ is constant as well, a contradiction. □
As well known S + m 1 × S 1 and S 1 are homotopically equivalent. Therefore a continuous map ϕ : M S + m 1 × S 1 is null homotopic if and only if π 2 ϕ : M S 1 is null homotopic. The homotopy classes of continuous maps M S 1 form an abelian group π 1 ( M ) (the Bruschlinski group of M) naturally isomorphic to H 1 ( M , Z ) . We may conclude that
Corollary 1.
Let M be a compact, orientable, connected Riemannian manifold with H 1 ( M , Z ) = 0 . Then every nonconstant exponentially harmonic map ϕ : M S m meets Σ.

Author Contributions

The two authors have equally contributed to the findings in the present work.

Funding

This research received no external funding.

Acknowledgments

Sorin Dragomir acknowledges support from Italian PRIN 2015. Francesco Esposito is grateful for support within the joint Ph.D. program of Università degli Studi della Basilicata (Potenza, Italy) and Università del Salento (Lecce, Italy).

Conflicts of Interest

The authors declare no conflict of interest.

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