Existence and Stability of Harmonic Maps
In this paper, we first study the energy functional, Euler-Lagrange operator, and -stress-energy tensor. Second, it is shown that the critical points of the energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal harmonic maps are minimal submanifolds. Then, the stability of any harmonic map on Riemannian manifold with nonpositive curvature is studied. Finally, the instability of harmonic maps from a compact manifold to a standard unit sphere is investigated.
Eells and Sampson introduced the concept of harmonic maps between Riemannian manifolds . If be a smooth map between Riemannian manifolds, then the energy functional of is denoted by and defined as follows:where is the volume element of . The Euler-Lagrange equation corresponding to is given bywhere is the induced connection on the pull-back bundle . The section described in (2) is known as the tension field of .
The authors showed in  that any smooth map from a compact Riemannian manifold to a Riemannian manifold with nonpositive sectional curvature can be deformed into a harmonic map. This is well-known as the fundamental existence theorem for harmonic maps. Following that, several researchers conducted a study on these maps [2–6]. Harmonic maps have been examined in numerous theories in mathematical physics, such as liquid crystal, ferromagnetic material, superconductors [7, 8].
Sacks and Uhlenbeck introduced perturbed energy functional that satisfies the Palais–Smale condition in their pioneering paper  in 1981 to prove the existence of harmonic maps from a closed surface and thus obtained so-called -harmonic maps, as critical points of perturbed functional, to approximate harmonic maps. For , the energy functional of the map is denoted by and defined as follows:where denotes the Hilbert–Schmidt norm of the differential map with respect to and . Noting that, for , satisfies Morse theory and Ljusternik–Schnirelman theory . Furthermore, the critical points of are said to be the -harmonic maps. Sacks and Uhlenbeck developed an existence theory for harmonic mappings of orientable surfaces into Riemannian manifolds using the critical maps of . Furthermore, they demonstrated that the convergence of the critical points of is sufficient to construct at least one harmonic map of the sphere into any Riemannian manifold . Many studies have recently been conducted on these maps. For example, the authors of  investigated the convergence behaviour of a sequence of harmonic mappings with from a compact surface into a compact Riemannian manifold without boundary. It is worth noticing that this sequence converges weakly to a harmonic map. Furthermore, in , it is studied the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up.
Following the concepts of [1, 4, 8–14], we study the stability, existence, and structure of -harmonic maps, as well as their practical applications, in this work. The existence of harmonic maps in an arbitrary class of homotopy and the conditions under which the fibers of harmonic maps are minimal submanifolds are explored in particular. Furthermore, the stability of harmonic mappings from a Riemannian manifold to a nonpositive Riemannian curvature and unit standard sphere is explored. Sections 3–4.1 present our key findings. This manuscript is organized as follows: Section 1 investigates the concepts of ɑ–energy functional and ɑ–harmonic maps, and it provides an explicit relationship between ɑ–harmonic maps and harmonic maps via conformal deformation. Section 2 looks at the existence of ɑ–harmonic mappings. It is demonstrated that, under certain assumptions, every smooth map between Riemannian manifolds can be deformed into an ɑ–harmonic map. The ɑ–stress energy tensor and its practical applications are discussed in Section 3. It is also shown that the vanishing of the divergence of the ɑ–stress energy tensor of ψ is identical to the vanishing of the ɑ–hormonicity of ψ. Furthemore, the criteria that cause the fibres of horizontally conformal ɑ–harmonic mappings to be minimal submanifolds are then examined. In Section 4, the Jacobi operator and Green’s theorem are used to derive the second variation formula of the ɑ–energy functional. The stability of ɑ–harmonic maps is explored in the last part. Any ɑ–harmonic map from a Riemannian manifold to a Riemannian manifold with nonpositive curvature is proved to be stable in this sense. The instability of ɑ–harmonic mappings from a compact manifold to a unit standard sphere is also explored.
1.1. -Harmonic Maps
In this section, first, we study the notion of harmonic maps. Then, the certain relation between -harmonic maps and harmonic maps through conformal deformation is given.
Let be a smooth map between Riemannian manifolds. Throughout this paper, it is considered that is a compact Riemannian manifold. Furthermore, the Levi–Civita connections on and are denoted by and , respectively. Moreover, the induced connection on the pullback bundle is denoted by and defined by , for any smooth vector field and section .
Let be a positive constant such that . Then, the energy functional of is defined as follows:
The critical points of are called harmonic maps. By Green’s theorem, the corresponding Euler-Lagrange equation of the energy functional, , is given by
The section is said to be the tension field of .
Theorem 1. Let be a smooth map between Riemannian manifolds. Then, is harmonic if and only if it has a vanishing tension field.
Definition 1. Let be a smooth map between Riemannian manifolds. It is called nondegenerate if the induced tangent map is nondegenerate, i.e.,
Example 1. Let and be a smooth map defined as follows:By equation (5) and Definition 1, it can be seen that is a nondegenerate harmonic map.
Now, an explicit relation between harmonic maps and harmonic maps through conformal deformation is given, which provides examples of [, Proposition 2.2], where .
Proposition 1. Let be a nondegenerate smooth map with . Then, is an harmonic map if and only if the map is harmonic with respect to the conformally related metric defined by
Proof 1. Assume that be a Riemannian metric for some smooth positive function on conformally associated to by . Denote the tension fields of the smooth map with respect to and by and , respectively. By (2), it can be obtained thatSettingBy (5), (7), and (8), we getThe Proposition 1 follows from (10).
2. The Existence of Harmonic Map
In this section, it is assumed that and are compact Riemannian manifolds, and is a homotopy class of a smooth given map . The following theorem is due to Hong.
Theorem 2 (see ). Suppose that is a harmonic map. Then, for any , there exists a smooth metric conformally equivalent to such that is harmonic and , where is the Hilbert–Schmidt norm with respect to and .
Theorem 3. Let and be a harmonic map. Then, there is a smooth metric on conformally equivalent to such that is harmonic if .
Proof 2. Assume that be a positive constant and . Let be the inverse function of on . Then, we haveon . SettingThe derivative of with respect to can be obtained as follows:Due to the fact that is the inverse function of and , we getfor . Then, on between and . Thus, there exists a smooth function on for some positive constant , such thatMoreover, by (11), we haveBy using (16) and (17), we getSince is a positive function on , by (18), it can be seen that is a positive function on .
Now, by Theorem 5, for , there exists a smooth metric conformally equivalent to such that is harmonic and . Due to the harmonicity of , the tension field of associated to the metrics and vanishes, . Assume that for a smooth positive function , by (8), we getDue to the fact that , the positive function can be defined as follows:By (18) and (20), it yields thatBy substituting (21) in (19) and using (5), we getAnd this completes the proof.
2.1. The Stress-Energy Tensor
Let be a smooth map between Riemannian manifolds. The stress-energy tensor of associated with the energy functional (in the sequel, we call the -stress-energy tensor of , in short) is denoted by and defined as follows:
Remark 1. The stress-energy tensor sometimes called the stress-energy-momentum tensor or the energy-momentum tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and nongravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity .
In general relativity, the symmetric stress-energy tensor acts as the source of spacetime curvature and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations .
The following Proposition 2 and Remark 2 provide examples of [, Proposition 4.1], in which and .
Proposition 2. Based on the above notations, we getfor all .
Proof 3. Choose a local orthonormal frame on with , at a point . Let be a smooth vector field on . According to the definition of divergence operator on a Riemannian manifold, at point , we haveAt point , the last term of the right-hand side of 24 can be obtained as follows:where we useat the point , for the third equality. Furthermore,By substituting (26) and (28) in the last equation of (25), we getand hence completes the proof.
Remark 2. Based on Proposition 2, it can be seen that if is a submersion and , then is an harmonic map. Conversely, if is an harmonic map, then .
3. Horizontal Conformal Harmonic Maps
Let be a smooth map between Riemannian manifolds and let for each . The vertical space at with respect to is denoted by and defined by . The orthogonal complement of in is denoted by and called the horizontal space at . For any , the tangent vector can be decomposed as follows:where and . The map is called a horizontally conformal map if there exists a positive function such thatfor any and . The function is said to be the dilation of .
Remark 3. In Relativistic Astrophysics, conformal mappings are used as a mathematical mechanism to obtain exact solutions to the Einstein field equations in general relativity. The behaviour of the space-time geometry quantities is given under a conformal transformation, and the Einstein field equations are exhibited for a perfect fluid distribution matter configuration .
The following Theorem 4 and Corollary 1 provide examples of [, Theorem 5.1].
Theorem 4. Let be a horizontally conformal harmonic map with dilation where and . Then, the fibers of are minimal submanifold if and only if is vertical.
Proof 4. For , choose a local orthonormal frame field near , such that are horizontal vectors and are vertical tangent vectors. Due to the fact that is horizontally conformal with dilation , we get . By (23), it is obtained thatBy Proposition 2 and considering the fact that the map is harmonic, we getFor , we haveBy making use of (33) and (34), for , we getBy (35) and the definition of the mean curvature vector , of the fiber of , given as follows:it is obtained thatFrom (37) and the assumptions of this theorem, it can be seen thatFrom (37) and (38), Theorem 4 follows.
Corollary 1. Under the assumptions of Theorem 4, the fibers of are minimal submanifold if and only if the horizontal distribution has mean curvature .
Proof 5. By (33), for , we getFor , it is obtained thatBy (39) and (40), we getThus, the mean curvature , of the horizontal distribution, is obtained as follows:From Theorem 4 and equation (42), Corollary 1 follows.
4. Second Variational Formula
In this section, the second variation formula of the energy functional is calculated by using the Jacobi operator and Stokes’ theorem, illuminating [, Theorem 6.1] in which .
Remark 4. In physics, the Jacobi operator plays a key role to solve the nonlinear boundary value of Troesch’s equation. This equation arises in the investigation of the confinement of a plasma column by radiation pressure and also in the theory of gas porous electrodes [14, 15].
Theorem 5. (The second variation formula) Let be an harmonic map and be a 2-parameter smooth variation of such that Then,whereand is given bywhere is the curvature tensor on , and denotes the inner product on
Proof 6. Let be a product manifold that is equipped with the product metric, and let the natural extension of on , on , and on to the product manifold is denoted by , and again, respectively. Assume that be a smooth map defined by . The same notation shall be used for the induced connection on . Consider an orthonormal frame with respect to on . Then, by (4), we haveBy computing the second derivative, we getwhere . Due to the fact thatwhere , and the first term of the right-hand side of (47) can be obtained as follows:Furthermore, by calculating the second term of the right-hand side of (47), we getBy the definition and properties of the curvature tensor of and the compatibility of with the metric , the last term of the right-hand side of (47) can be obtained as follows:From equations (46), (47), (49), (50), and (44) and considering Green’s theorem and harmonicity of , Theorem 5 follows.
4.1. Stability of -Harmonic Maps
In this section, the stability of harmonic maps is studied. First, we showed that any harmonic map from a Riemannian manifold to a Riemannian manifold with nonpositive curvature is stable. Then, the stability of harmonic maps from a compact manifold to a unit standard sphere is investigated.
Remark 5. The stability of harmonic maps plays a key role in mathematical physics and mechanics, . For instance, the linearized Vlasov–Maxwell equations are used to investigate harmonic stability properties for planar wiggler free-electron laser (FEL). It is worth noting that the analysis is carried out in the Compton regime for a tenuous electron beam propagating in the direction through the constant amplitude planar wiggler magnetic field .
Definition 2. Under the assumptions of Theorem 5, settingThen, is said to be stable harmonic if for any compactly supported vector field along .
From Theorem 5, we obtain the following corollary and provide examples of [, Theorem 6.2], in which harmonic map is harmonic map.
Corollary 2. Let be a Riemannian manifold with nonpositive Riemannian curvature. Then, any harmonic map is stable.
Proof 7. Settingfor any . Then, we haveOn the other hand, by (54), we getwhere . By substituting (55) and (56) in (45) and using the Divergence theorem and Definition 2, we getBy (57) and the assumptions, Corollary 2 follows.
Now, the stability of harmonic maps from a compact without boundary Riemannian manifold to a standard unit sphere is studied. As in , an extrinsic average variational method of Wei ([17, 18]) is employed.
Consider a unit standard sphere as a submanifold in dimensional Euclidean space . Denote the Levi-Civita connections on and by and , respectively. At , any vector in can be decomposed as follows:where is the normal part to , and is the tangent part to . The second fundamental form of in is denoted by and defined as follows:where are tangent vectors of at and is the Euclidean metric on . Furthermore, the shape operator corresponding to a normal vector field on is defined bywhere is a tangent vector field on . Noting that, the second fundamental form and the shape operator of are satisfied by the following equation:for any tangent vectors and of at .
From the sectional curvature of and using (59), (60), and (61), the following lemma is obtained.
Lemma 1. Let be a smooth map, and let is a parallel vector field in . Then, at , we have(1),(2),(3),(4),where and denote the induced connection on and the curvature tensor of , respectively.
Theorem 6. Let be a compact without boundary manifold and let be an harmonic map such that . Then, is unstable.
Proof 9. Let be a parallel orthonormal frame field in and be a local orthonormal frame field on , and let and denote the curvature tensor of and the induced connection on , respectively. By (57), we havewhere . According to Lemma 1’s second assertion, the first term of the right-hand side of (65) can be obtained as follows:By virtue of Lemma 1’s third claim, the second term of the right-hand side of (65) can be calculated as follows:Since the sectional curvature of is constant and using Lemma 1’s fourth assertion, the last term of the right-hand side of (65) can be calculated as follows:By substituting (66), (67), and (68) in (65), we getwhere . By (69) and the assumptions, it is obtained thatThen, is unstable and hence completes the proof.
No data were used to support this study.
An earlier version of the manuscript has been presented in  as a preprint.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
S. Tanveer, “Singularities in the classical Rayleigh-Taylor flow: formation and subsequent motion,” in Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, vol. 441, pp. 501–525, London, UK, 1993.View at: Google Scholar
S. M. Kazemi Torbaghan and M. M. Rezaii, “Stability of F-biharmonic maps,” Bulletin of the Iranian Mathematical Society, vol. 43, no. 6, pp. 1657–1669, 2017.View at: Google Scholar
Z. Pirbodaghi, M. Mirmohammad Rezaii, and S. M. Kazemi Torbaghan, “Stability of λ-harmonic maps,” Mathematics, vol. 6, no. 94, 2018.View at: Google Scholar
J. Sacks and K. Uhlenbeck, “The existence of minimal immersions of 2-spheres,” Annals of Mathematics, pp. 1–24, 1981.View at: Google Scholar
J. Li and X. Zhu, “Energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps to a sphere,” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, Elsevier Masson, Paris, France, vol. 36, no. 1, pp. 103–118, 2019.View at: Google Scholar
Y. Li and Y. Wang, “A weak energy identity and the length of necks for a sequence of Sacks–Uhlenbeck α-harmonic maps,” Advances in Mathematics, vol. 225, no. 3, pp. 1134–1184, 2010.View at: Google Scholar
M. Ara, “Geometry of F-harmonic maps,” Kodai Mathematical Journal, vol. 22, no. 2, pp. 243–263, 1999.View at: Google Scholar
N. E. Djaa and F. Latti, “Geometry of generalized F-harmonic maps,” Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science, pp. 101–114, 2020.View at: Google Scholar
V. S. Markin, A. A. Chernenko, Y. A. Chizmadehev, and Y. G. Chirkov, “Aspects of the theory of gas porous electrodes,” Fuel Cells: Their Electrochemical Kinetics, Consultants Bureau, New York, NY, USA, pp. 22–33, 1966.View at: Google Scholar
S. W. Wei, “An average process in the calculus of variations and the stability of harmonic maps,” Bulletin of the Institute of Mathematics, Academia Sinica, vol. 11, no. 3, pp. 469–474, 1983.View at: Google Scholar
S. W. Wei, An Extrinsic Average Variational Method, Recent Developments in Geometry (Los Angeles, CA, Contemporary Mathematics, American Mathematical Society, Providence, RI, USA, 1987.