Exponentially Harmonic Maps into Spheres

: 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 (cid:54) = ∅ .

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) (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).

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 where t =t • Π 2 ,t is the Cartesian coordinate on R, and 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 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 Also for an arbitrary test function ϕ ∈ C ∞ (M) we set for every tangent vector field X ∈ 4 on U . On the other hand Hence (differentiating with respect to s) is the divergence operator with respect to the Riemannian volume form (5) into (4) together with Green's lemma yields [by {dE(φ s )/ds} s=0 = 0 and the density of Equation (7) is part of the Euler-Lagrange system associated to the variational principle δ E(φ) = 0. Next [by (7)] Let us integrate over M in (8) and use Green's lemma. We obtain yielding [as φ is assumed to be nonconstant] u(x) = t φ for some t φ ∈ R and any x ∈ M . Q.e.d.
Hence (differentiating with respect to s) where div: 4 on U . On the other hand Hence (differentiating with respect to s) where div : X(M ) → C ∞ (M ) is the divergence operator with respect to the Riemannian volume form i.e. L X d v g = div(X) d v g and ∆ is the Laplace-Beltrami operator (on functions) i.e. ∆u = −div(∇u). Substitution from (5) Equation (7) is part of the Euler-Lagrange system associated to the variational principle δ E(φ) = 0. Next [by (7)] Let us integrate over M in (8) and use Green's lemma. We obtain yielding [as φ is assumed to be nonconstant] u(x) = t φ for some t φ ∈ R and any x ∈ M . Q.e.d.
(M) → C ∞ (M) is the divergence operator with respect to the Riemannian volume form i.e., L X d v g = div(X) d v g and ∆ is the Laplace-Beltrami operator (on functions) i.e., ∆u = −div(∇u). Substitution from (5) into (4) together with Green's lemma yields [by {dE(φ s )/ds} s=0 = 0 and the density of which is (2) in Lemma 1. When w t = 0 Equation (6) is Equation (7) is part of the Euler-Lagrange system associated to the variational principle δ E(φ) = 0. Next (by (7) 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.

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 : 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 Let us consider the map Let g N denote the canonical Riemannian metric on S N ⊂ R N+1 . The map I is an isometry of S m−1 Let us consider the mapψ = 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|< of f 0 = f we set f s = π • f s so that { f s } |s|< 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| < . Q.e.d.
By Lemma 2 the mapψ = I −1 • φ is exponentially harmonic. Let us set where π 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 =ũ(x 0 ) ∈ S 1 . Also, considered the covering map p : R → S 1 , p(t) = exp(2πit), pick t 0 ∈ R such that p(t 0 ) = ζ 0 . As φ is null homotopic, the mapψ is null homotopic as well. Theñ 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 is exponentially harmonic. We shall close the proof of Theorem 1 by showing that exponentially harmonic mappings into S m−1 + are constant.

Exponentially Harmonic Map System in Divergence Form
Let us consider the L 2 inner products 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 us substitute from (13) into (1) [with φ i = F i ] and take into account The exponentially harmonic map system (1) becomes Multiplication of (15) by exp e(F) and subtraction from (9) [with u = F i ] yields (10) for any 1 ≤ i ≤ m − 1.
We may now end the proof of Theorem 1 as follows. Let