axioms

Article Exponentially Harmonic Maps into Spheres

Sorin Dragomir 1,* and Francesco Esposito 2

1 Dipartimento di Matematica, Informatica, ed Economia, Universita` degli Studi della Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy 2 Dipartimento di Matematica e Fisica Ennio De Giorgi, Universita` del Salento, 73100 Lecce, Italy; [email protected] * Correspondence: [email protected]



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 Sm Rm 1. Given a codimension two totally geodesic ⊂ submanifold Σ Sm, we show that every nonconstant exponentially harmonic map φ : M Sm ⊂ → either meets or links Σ. If H1(M, Z) = 0 then φ(M) Σ = ∅. ∩ 6 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). → 1/2 The Hilbert-Schmidt norm of dφ is dφ = traceg (φ∗h) : M R. Let us consider the functional k k →   ∞ 1 2 E : C (M, N) R, E(φ) = exp dφ d vg . → M 2 k k Z   ∞ A C map φ : M N is exponentially harmonic if it is a critical point of E i.e., d E(φs)/ds s=0 = 0 → ∞ { } for any smooth 1-parameter variation φs s

d φ E(φs) = = exp e(φ) h (V, τ(φ) + φ e(φ)) d vg ds s 0 ∗ { } − ZM ∇   where e(φ) = 1 dφ 2 and τ(φ) C∞ φ 1TN is the tension field of φ (cf. e.g., [2]). Also 2 k k ∈ − V = ∂φ /∂s is the infinitesimal variation induced by the given 1-parameter variation. s s=0
In particular, the Euler-Lagrange equations of the variational principle δ E(φ) = 0 are
∂φj ∂φk ∂φi ∂e(φ) ∆φi + Γi φ gαβ + gαβ = 0 (1) − jk ◦ ∂xα ∂xβ ∂xα ∂xβ   where 1 ∂ ∂u ∆u = √G gαβ , G = det[g ], − √ ∂xα ∂xβ αβ G   i is the Laplace-Beltrami operator and Γjk are the Christoffel symbols of hij. 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 = Sm. 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]).

Axioms 2018, 7, 88; doi:10.3390/axioms7040088 www.mdpi.com/journal/axioms Axioms 2018, 7, 88 2 of 7

The purpose of the present paper is to further study exponentially harmonic maps φ winding in N = Sm, a situation previously attacked in [4], though confined to the case where M is a Fefferman m 1 space-time (cf. [9]) over the Heisenberg group Hn and φ : M S is S invariant. Fefferman spaces → are Lorentzian manifolds and exponentially harmonic maps of this sort are usually referred to as m exponential wave maps (cf. e.g., Y-J. Chiang & Y-H. Yang, [10]). Base maps f : Hn S associated → (by the S1 invariance) to φ : M Sm 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 Sm 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 Sm from a compact Riemannian manifold either φ(M) Σ = ∅ → ∩ 6 or φ : M Sm Σ isn’t homotopically null. Here Σ Sm 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

i i ∗ exp e(φ) φ + 2 e(φ) exp e(φ) φ = 0 −∇ ∇       (got by a verbatim repetition of arguments in [4]), (ii) observing that Sm Σ is isometric to the warped m 1 1 \ product manifold S − w S , 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 gL. Let w C (S) × ∈ such that w(y) > 0 for any y S and let us endow S with the warped product metric ∈ 2 h = Π∗ g + w dt dt, 1 L ⊗ where t = t˜ Π2, t˜ is the Cartesian coordinate on R, and ◦

Π1 : S L, Π2 : 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 φ, u = Π φ. 1 ◦ 2 ◦ 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 w φ ∆u + φ u 2 (2) ◦ ∂t ◦ k∇ k  
= w φ ( u) e(φ) + 2 ( u)(w φ). ◦ ∇ ∇ ◦ If additionally ∂w/∂t = 0 then φ (M)
L tφ for some tφ R. ⊂ × { } ∈ 4

Axioms 2018, 7, 88 on U. On the other hand 3 of 7 2 2 (3) φ∗h (X,X) = F ∗g (X,X) + (w φ ) X(u) + s X(ϕ) s L ◦ s Also for an arbitraryfortest every function tangent
ϕ vectorC∞( fieldM
) weX setX(M). Formula (3) for X = Eα yields ∈ ∈ 2 dφφ (x2)= = dFF(x2),+u(wx) +φ s ϕ(x) u, 2x+ 2sM g,( us , <ϕe), + s2 ϕ 2 . k s sk k k ◦ s k∇ k ∈ ∇| | ∇ k∇ k Hence (differentiating with respect to s)

 ∞ so that φs s

div exp e(φ) (w φ)2 u = 0. (7) ◦ ∇ n   o Equation (7) is part of the Euler-Lagrange system associated to the variational principle δ E(φ) = 0. Next (by (7)) 2 2 div w φ u exp e(φ) u = exp e(φ) w φ u 2 . (8) ◦ ∇ ◦ k∇ k n o Let us integrate over M in
(8) and use Green’s lemma. We obtain

2 2 exp e(φ) w φ u d vg = 0 ZM ◦ k∇ k  
Axioms 2018, 7, 88 4 of 7

yielding (as φ is assumed to be nonconstant) u(x) = tφ for some tφ R and any x M. Q.e.d. ∈ ∈ 3. Exponentially Harmonic Maps Omitting a Codimension 2 Sphere Aren’t Null Homotopic Let Σ Sm be a codimension 2 totally geodesic submanifold. A continuous map φ : M Sm ⊂ → meets Σ if φ(M) Σ = ∅ and links Σ if φ(M) Σ = ∅ and φ : M Sm Σ is not null homotopic. ∩ 6 ∩ → \ The purpose of the section is to establish

Theorem 1. Let φ : M Sm be a nonconstant exponentially harmonic map from a compact, connected, → + orientable Riemannian manifold M into the sphere Sm Rm 1. If Σ Sm 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 Sm Σ m+1 → \ is null homotpic. Let (ξj) be a system of coordinates on R such that Σ is given by the equations m 1 m ξ1 = ξ2 = 0. Let S+− R be the hemisphere ⊂

m 1 m 1 m 1 S+− = y = (y0 , ym) R − R : y S − , ym > 0 . ∈ × ∈ n o Let us consider the map

m 1 1 m I : S − S S Σ, I(y, ζ) = y u, y v, y0 , + × → \ m m m 1 1
y = y0 , ym S+− , ζ = u + i v S C. ∈ ∈ ⊂ N N+1 Let gN denote the canonical Riemannian
metric on S R . The map I is an isometry of m 1 1 m ⊂ S − S onto (S Σ , gm) with the warping function + × f \ ∞ m 1 1 f C (S − S ), 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 fs s

u˜ π1(M, x0) = 0 ∗ where π1(M, x0) is the first homotopy group of M. Consequently there is a unique smooth function u : M R such that p u = u˜ and u(x0) = t0. The map → ◦ m 1 ψ = (F, u) : M S+− w R → × Axioms 2018, 7, 88 5 of 7 is exponentially harmonic [because ψ = π ψ˜ and ◦ m 1 m 1 1 π = 1 m 1 , p : S+− w R S+− f S S+− × → ×   ∞ m 1 is a local isometry, where w C (S+− ) is given by w(y) = ym]. We may then apply Lemma1 to the m 1 ∈ map ψ with L = S+− to conclude that

m 1 ψ(M) S − t ⊂ + × ψ m 1  for some tψ R. It follows that F = π1 ψ : M S+− is exponentially harmonic. We shall close the ∈ ◦ → m 1 proof of Theorem1 by showing that exponentially harmonic mappings into S+− are constant.

4. Exponentially Harmonic Map System in Divergence Form Let us consider the L2 inner products

(u, v)L2 = uv d vg , (X, Y)L2 = g(X, Y) d vg . ZM ZM Let us think of the gradient as a first order differential operator : C1 M) C T(M) and ∇ ∇ → let be its formal adjoint, i.e., ∇∗
∗X , u = X , u ∇ L2 ∇ L2 for any X C1 T(M) and u C1(M). Ordinary
integration
by parts shows that X = div(X). ∈ ∈ ∇∗ − Let ρ = exp e(F) C∞(M). Starting from ∆u = div( u) one has ∈
− ∇   ρ ∆u, ϕ = ∗ u, ρϕ = u, (ρϕ) L2 ∇ ∇ L2 ∇ ∇ L2


= ( ) + ( ) ∗ ρ u , ϕ L2 ϕ g u, ρ d vg ∇ ∇ ZM ∇ ∇ for any ϕ C∞(M), that is
∈ exp e(F) ∆u = ∗ exp e(F) u (9) ∇ ∇   + exp e(F) g  u,
e(F) . ∇ ∇ m 1  
m 1 m Lemma 3. Let F : M S+− be an exponentially harmonic map and F = j F where j : S − , R is the → ◦ → inclusion. If F = F1 , , Fm then ···
i i ∗ exp e(F) F + 2 e(F) exp e(F) F = 0 (10) − ∇ ∇       for any 1 i m. ≤ ≤ 1 m 1 m 1 m 1 m 1 m 1 Proof. Let y = y , , y − : S+− B − be the projection, where B − R − is the open ··· → ⊂ unit ball. With respect to this choice of local coordinates, the standard metric gm 1 and its Christoffel
− symbols are yiyj m 1 = + 2 = − i 2 hij δij 2 , y ∑ y , (11) 1 y | | i=1 − | |
hij = δij yiyj , (12) − i i Γjk = y hjk . (13) Axioms 2018, 7, 88 6 of 7

Let us substitute from (13) into (1) [with φi = Fi] and take into account

1 ∂Fj ∂Fk e(F) = gαβ h F . (14) 2 ∂xα ∂xβ jk ◦
The exponentially harmonic map system (1) becomes

∆Fi + 2 e(F) Fi + g e(F), Fi = 0, 1 i m 1. (15) − ∇ ∇ ≤ ≤ −
Multiplication of (15) by exp e(F) and subtraction from (9) [with u = Fi] 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 m 2 m 1 i 2 (F ) = 1 ∑ − (F ) together with (11) and (14) to show that − i=1 m 1 2 e(F) = ∑ Fj . 2 j=1 ∇

Finally, one contracts (15) by Fi and uses once again the constraint together with ∆(u2) = 2 u ∆u u 2 . Q.e.d. − k∇ k m 1 We may now end the proof of Theorem1 as follows. Let F : M S − 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 d vg = 0 ZM   and Fm > 0 so that 1 m 0 = e(F) = ∑ Fj 2 2 j=1 k∇ k yielding Fj = constant. So φ is constant as well, a contradiction.

m 1 1 1 As well known S+− S and S are homotopically equivalent. Therefore a continuous map m 1 1 × 1 φ : M S − S is null homotopic if and only if π φ : M S is null homotopic. The homotopy → + × 2 ◦ → classes of continuous maps M S1 form an abelian group π1(M) (the Bruschlinski group of M) → naturally isomorphic to H1(M, Z). We may conclude that

Corollary 1. Let M be a compact, orientable, connected Riemannian manifold with H1(M, Z) = 0. Then every nonconstant exponentially harmonic map φ : M Sm 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 Universita` degli Studi della Basilicata (Potenza, Italy) and Universita` del Salento (Lecce, Italy). Conflicts of Interest: The authors declare no conflict of interest.

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