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Axioms 2018, 7(4), 88; doi:10.3390/axioms7040088
Exponentially Harmonic Maps into Spheres
Dipartimento di Matematica, Informatica, ed Economia, Università degli Studi della Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy
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
We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere . Given a codimension two totally geodesic submanifold , we show that every nonconstant exponentially harmonic map either meets or links . If then .
Keywords:exponentially harmonic map; totally geodesic submanifold; Euler-Lagrange equations
Let M be a compact, connected, orientable n-dimensional Riemannian manifold, with the Riemannian metric g. Let be a map into another Riemannian manifold . The Hilbert-Schmidt norm of is . Let us consider the functional
A map is exponentially harmonic if it is a critical point of E i.e., for any smooth 1-parameter variation of . Exponentially harmonic maps were first studied by J. Eells & L. Lemaire , who derived the first variation formulawhere and is the tension field of (cf. e.g., ). Also is the infinitesimal variation induced by the given 1-parameter variation. In particular, the Euler-Lagrange equations of the variational principle arewhereis the Laplace-Beltrami operator and are the Christoffel symbols of . The (partial) regularity of weak solutions to (1) was investigated by D.M. Duc & J. Eells (cf. ) when and by Y-J. Chiang et al. (cf. ) when . Differential geometric properties of exponentially harmonic maps, including the second variation formula for E, were found by M-C. Hong (cf. ), J-Q. Hong & Y. Yang (cf. ), L-F. Cheung & P-F. Leung (cf. ), and Y-J. Chiang (cf. ).
The purpose of the present paper is to further study exponentially harmonic maps winding in , a situation previously attacked in , though confined to the case where M is a Fefferman space-time (cf. ) over the Heisenberg group and is 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, ). Base maps associated (by the invariance) to turn out to be solutions to degenerate elliptic equations [resembling (cf. ) the exponentially harmonic map system (1)] and the main result in  is got by applying regularity theory within subelliptic theory (cf. e.g., ).
Through this paper, M will be a compact Riemannian manifold and 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, ), we succeed in recovering, to the setting of exponentially harmonic maps, the result by B. Solomon (cf. ) that for any nonconstant harmonic map from a compact Riemannian manifold either or isn’t homotopically null. Here is an arbitrary codimension 2 totally geodesic submanifold.
The ingredients in the proof of the exponentially harmonic analog to Solomon’s theorem (see ) are (i) setting the Equation (1) in divergence form(got by a verbatim repetition of arguments in ), (ii) observing that is isometric to the warped product manifold , 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 , where L is a Riemannian manifold with the Riemannian metric . Let such that for any and let us endow S with the warped product metricwhere , is the Cartesian coordinate on , andare projections. The Riemannian manifold is customarily denoted by . Let be an exponentially harmonic map and let us set
We need to establish the following
Let M be a compact, connected, orientable Riemannian manifold anda nonconstant exponentially harmonic map. Then u is a solution to
If additionally then for some .
Also for an arbitrary test function we setso that is a 1-parameter variation of . For each let be a local g-orthonormal (i.e., ) frame, defined on an open neighborhood of . Thenon U. On the other handfor every tangent vector field . Formula (3) for yields
Hence (differentiating with respect to s)where . Moreoverwhere is the divergence operator with respect to the Riemannian volume formi.e., and is the Laplace-Beltrami operator (on functions) i.e., . Substitution from (5) into (4) together with Green’s lemma yields [by and the density of in ]which is (2) in Lemma 1. When Equation (6) is
Equation (7) is part of the Euler-Lagrange system associated to the variational principle . Next (by (7))
Let us integrate over M in (8) and use Green’s lemma. We obtainyielding (as is assumed to be nonconstant) for some and any . Q.e.d.
3. Exponentially Harmonic Maps Omitting a Codimension 2 Sphere Aren’t Null Homotopic
Let be a codimension 2 totally geodesic submanifold. A continuous map meets if and links if and is not null homotopic. The purpose of the section is to establish
Let be a nonconstant exponentially harmonic map from a compact, connected, orientable Riemannian manifold M into the sphere . If is a codimension 2 totally geodesic submanifold, then φ either meets or links Σ.
The proof is by contradiction, i.e., we assume that doesn’t meet and the map is null homotpic. Let be a system of coordinates on such that is given by the equations . Let be the hemisphere
Let us consider the map
Let denote the canonical Riemannian metric on . The map I is an isometry of onto with the warping function
Let us consider the map . We need the following. □
Let S andbe Riemannian manifolds,a local isometry, andan exponentially harmonic map. Then every mapsuch thatis exponentially harmonic.
Let h and be the Riemannian metrics on S and . For every 1-parameter variation of we set so that is a 1-parameter variation of . A calculation relying on yields for every . Q.e.d.
By Lemma 2 the map is exponentially harmonic. Let us setwhere and are projections. Next let us consider a point and set . Also, considered the covering map , , pick such that . As is null homotopic, the map is null homotopic as well. Thenwhere is the first homotopy group of M. Consequently there is a unique smooth function such that and . The mapis exponentially harmonic [because andis a local isometry, where is given by ]. We may then apply Lemma 1 to the map with to conclude thatfor some . It follows that is exponentially harmonic. We shall close the proof of Theorem 1 by showing that exponentially harmonic mappings into are constant. □
4. Exponentially Harmonic Map System in Divergence Form
Let us consider the inner products
Let us think of the gradient ∇ as a first order differential operator and let be its formal adjoint, i.e.,for any and . Ordinary integration by parts shows that . Let . Starting from one hasfor any , that is
Letbe an exponentially harmonic map andwhereis the inclusion. Ifthenfor any.
Let be the projection, where is the open unit ball. With respect to this choice of local coordinates, the standard metric and its Christoffel symbols are
The exponentially harmonic map system (1) becomes
To see that (15) (and therefore (10)) holds for as well, one first exploits the constraint together with (11) and (14) to show that
Finally, one contracts (15) by and uses once again the constraint together with . Q.e.d.
We may now end the proof of Theorem 1 as follows. Let be an exponentially harmonic map. Let us integrate over M in (10) for . Then (by Green’s lemma)and so thatyielding constant. So is constant as well, a contradiction. □
As well known and are homotopically equivalent. Therefore a continuous map is null homotopic if and only if is null homotopic. The homotopy classes of continuous maps form an abelian group (the Bruschlinski group of M) naturally isomorphic to . We may conclude that
Let M be a compact, orientable, connected Riemannian manifold with. Then every nonconstant exponentially harmonic mapmeets Σ.
The two authors have equally contributed to the findings in the present work.
This research received no external funding.
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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