International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 104274 | 14 pages | https://doi.org/10.1155/2009/104274

Biwave Maps into Manifolds

Academic Editor: Jie Xiao
Received08 Jan 2009
Accepted30 Mar 2009
Published29 Jun 2009

Abstract

We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if is a biwave map into a Riemannian manifold under certain circumstance, then is a wave map. We verify that if is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then is a wave map. We finally obtain a theorem involving an unstable biwave map.

1. Introduction

Harmonic maps between Riemannian manifolds were first introduced and established by Eells and Sampson [1] in 1964. Afterwards, there were two reports on harmonic maps by Eells and Lemaire [2, 3] in 1978 and 1988. Biharmonic maps, which generalized harmonic maps, were first studied by Jiang [4, 5] in 1986. In this decade, there has been progress in biharmonic maps made by Caddeo et al. [6, 7], Loubeau and Oniciuc [8], Montaldo and Oniciuc [9], Chiang and Wolak [10], Chiang and Sun [11, 12], Chang et al. [13], Wang [14, 15], and so forth.

Wave maps are harmonic maps on Minkowski spaces, and their equations are the second-order hyperbolic systems of partial differential equations, which are related to Einsteinโ€™s equations and Yang-Mills fields. In recent years, there have been many new developments involving local well-posedness and global-well posedness of wave maps into Riemannian manifolds achieved by Klainerman and Machedon [16, 17], Shatah and Struwe [18, 19], Tao [20, 21], Tataru [22, 23], and so forth. Furthermore, Nahmod et al. [24] also studied wave maps from into (compact) Lie groups or Riemannian symmetric spaces, that is, gauged wave maps when , and established global existence and uniqueness, provided that the initial data are small. Moreover, Chiang and Yang [25] , Chiang and Wolak [26] have investigated exponential wave maps and transversal wave maps.

Bi-Yang-Mills fields, which generalize Yang-Mills fields, have been introduced by Ichiyama et al. [27] recently. The following connection between bi-Yang Mills fields and biwave equations motivates one to study biwave maps.

Let be a principal fiber bundle over a manifold with structure group and canonical projection , and let be the Lie algebra of . A connection can be considered as a -valued 1-form locally. The curvature of the connection is given by the 2-form with The bi-Yang-Mills Lagrangian is defined where is the adjoint operator of the exterior differentiation on the space of -valued smooth forms on (, the endormorphisms of ). Then the Euler-Lagrange equation describing the critical point of (1.2) has the form which is the bi-Yang-Mills system. In particular, letting and , the group of orthogonal transformations on we have that is a skew symmetric matrix The appropriate equivariant ansatz has the form where is a spatially radial function. Setting and the bi-Yang-Mills system (1.3) becomes the following equation for : which is a linear nonhomogeneous biwave equation, where is a function of and .

Biwave maps are biharmonic maps on Minkowski spaces. It is interesting to study biwave maps since their equations are the fourth-order hyperbolic systems of partial differential equations, which generalize wave maps. This is the first attempt to study biwave maps and their relationship with wave maps. There are interesting and difficult problems involving local well posedness and global well posedness of biwave maps into Riemannian manifolds or Lie groups (or Riemannian symmetric spaces), that is, gauged biwave maps for future exploration.

In Section 2, we compute the first variation of the bi-energy functional of a biharmonic map using tensor technique, which is different but much easier than Jiangโ€™s [4] original computation. In Section 3, we prove in Theorem 3.3 that if โ€‰โ€‰is a biwave map and โ€‰โ€‰ is a totally geodesic map, then โ€‰โ€‰ is a biwave map. Then we can apply this theorem to provide many biwave maps (see Example 3.4). We also can construct biwave nonwave maps as follow: Let โ€‰โ€‰be a wave map on a compact domain and let โ€‰โ€‰ be an inclusion map. The map โ€‰โ€‰ is a biwave nonwave map if and only if โ€‰โ€‰ has constant energy density, compare with Theorem 3.5. Afterwards, we show that if โ€‰โ€‰is a biwave map on a compact domain into a Riemannian manifold satisfying then is a wave map (cf. Theorem 3.6). This theorem is different than the theorem obtained by Jiang [4]: if โ€‰โ€‰ is a biharmonic map from a compact manifold into a Riemannian manifold with nonpositive curvature, then โ€‰โ€‰ is a harmonic map. In Section 4, we verify that if โ€‰โ€‰ is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then โ€‰โ€‰ is a wave map (cf. Theorem 4.5). We also prove that if โ€‰โ€‰ is a wave map on a compact domain with constant energy density, then โ€‰โ€‰is an unstable biwave map (cf. Theorem 4.7).

2. Biharmonic Maps

A biharmonic map from an -dimensional Riemannian manifold into an -dimensional Riemannian manifold is the critical point of the bi-energy functional where is the volume form on .

Notations
is the adjoint of and is the tension field. Here is the Riemannian connection on induced by the Levi-Civita connections on and , and is the local frame at a point of . The tension field has components where and are the Christoffel symbols on and , respectively.

In order to compute the Euler-Lagrange equation of the bi-energy functional, we consider a one-parameter family of maps from a compact manifold (without boundary) into a Riemannian manifold . Here is the endpoint of a segment starting at (=), determined in length and direction by the vector field along For a nonclosed manifold , we assume that the compact support of is contained in the interior of (we need this assumption when we compute by applying the divergence theorem). Then we have

Let The components of are We can use the curvature formula on and get where is the Riemannian curvature of . But therefore, has components We can rewrite (2.3) as where is a linear equation for and is an operator from to Solutions of are called Jacobi fields. Hence, we obtain the following definition from (2.3), (2.5), and (2.6).

Definition 2.1. is a biharmonic map if and only if the bitension field that is, the tension field , is a Jacobi field.
If , then . Thus, harmonic maps are obviously biharmonic. Biharmonic maps satisfy the fourth-order elliptic systems of PDEs, which generalize harmonic maps. Our computation for the first variation of the bi-energy functional presented here using tensor technique is different but much easier than Jiangโ€™s [4] original computation (it took him four pages).
Caddeo et al. [7] showed that a biharmonic curve on a surface of nonpositive Gaussian curvature is a geodesic (i.e., is harmonic) and gave examples of biharmonic nonharmonic curves on spheres, ellipses, unduloids, and nodoids.

Theorem 2.2 (see [4]). Let be an isometric embedding of an -dimensional compact Riemannian manifold into an -dimensional unit sphere with nonzero constant mean curvature. The map is biharmonic if and only if where is the second fundamental form of

Example 2.3. In the compact hypersurfaces, whose Gauss maps are isometric embeddings, are the Clifford surfaces [28]: Let be a standard embedding such that Because and is a biharmonic nonharmonic map by Theorem 2.2.

3. Biwave Maps

Let be an dimensional Minkowski space with the metric and the coordinates and let be an -dimensional Riemannian manifold. A wave map is a harmonic map on the Minkowski space with the energy functional The Euler-Lagrange equation describing the critical point of (3.1) is where is the wave operator on and are the Christoffel symbols of . is a wave map iff the wave field (i.e., the tension field on a Minkowski space) vanishes. The wave map equation is invariant with respect to the dimensionless scaling But, the energy is scale invariant in dimension .

If is a smooth map from a Minkowski space into a Riemannian manifold , then the bi-energy functional is, from (2.1), The Euler-Lagrange equation describing the critical point of (3.3), from (2.5), is

Definition 3.1. from a Minkowski space into a Riemannian manifold is a biwave map if and only if the biwave field (i.e., the bitension field on a Minkowski space), that is, the wave field , is a Jacobi field on the Minkowski space.
Biwave maps satisfy the fourth-order hyperbolic systems of PDEs, which generalize wave maps. If , then . Waves maps are obviously biwave maps, but biwave maps are not necessarily wave maps.

Example 3.2. Let be a function defined on a Minkowski space satisfying the following conditions: where the initial data and are given. Since this is a fourth-order homogeneous linear biwave equation with constant coefficients, it is well known that can be solved by [18, 29].

Let be a smooth map from a Minkowski space into a Riemannian manifold and let be a smooth map between two Riemannian manifolds and . Then the composition is a smooth map. Since is a semi-Riemannian manifold (i.e., a pseudo-Riemannian manifold), we can define a Levi-Civita connection on by Oโ€™Neill [30]. Let , be the connections on , respectively, and let be the curvatures on , respectively. We first have the following two formulas: for and for

Theorem 3.3. If is a biwave map and is totally geodesic between two Riemannian manifolds and then the composition is a biwave map.

Proof. Let be the coordinates of a point in and let be the frame at . We know from [4] that Since is totally geodesic, we have by applying the chain rule of the wave field to as [1]. Then we get Recalling that we derive from (3.7a) that since is totally geodesic. Therefore, we have Substituting (3.10) into (3.8), we arrive at where
On the other hand, we have by (3.7b) We obtain from (3.11) and (3.12) that is, Hence, if is a biwave map and is totally geodesic, then is a biwave map. Note that the total geodesicity of cannot be weakened into a harmonic or biharmonic map.

Example 3.4. Let be a submanifold of . Are the biwave maps into also biwave maps into ? The answer is affirmative iff is a totally geodesic submanifold of , that is, geodesics are geodesics. is a geodesic with iff is parallel, that is, iff For a map letting we have by (3.13) the following: since is a geodesic. Hence, is a biwave map if and only if solves the fourth-order homogeneous linear biwave equation as in Example 3.2. It follows from Theorem 3.3 that there are many biwave maps provided by geodesics of .

We also can construct examples of biwave nonwave maps from some wave maps with constant energy using Theorem 3.5. Let be a hypersphere of Then is a biharmonic nonminimal submanifold of by Theorem 2.2 and Example 2.3. Let be a unit section of the normal bundle of in Then the second fundamental form of the inclusion is By computation, the tension field of is , and the bitension field is .

Theorem 3.5. Let be a nonconstant wave map on a compact space-time domain and let be an inclusion. The map is a biwave nonwave map if and only if has constant energy density .

Proof. Let be the coordinate of a point in and let be the frame at . Recall that is the unit section of the normal bundle. By applying the chain rule of the wave field to we have since is a wave map. We can derive the following at the point by straightforward calculation: Therefore, we obtain Suppose that is a biwave nonwave map (). As the -part of , vanishes, which implies that is constant since is compact. The converse is obvious.

Let be the coordinates of a point in a compact space-time domain and be the frame at the point. Suppose that is a biwave map from a compact domain into a Riemannian manifold such that the compact supports of and are contained in the interior of

Theorem 3.6. If is a biwave map from a compact domain into a Riemannian manifold such that then is a wave map.

Proof. Since is a biwave map, we have by (3.4) Recall that are the coordinates of a point in and We compute By applying the Bochnerโ€™s technique from (3.19) and the assumption that the compact supports of and are contained in the interior of we know that is constant, that is, . If we use the identity and the fact then we can conclude that by applying the divergence theorem.

Corollary 3.7. If is a biwave map on a compact domain such that and , then is a wave map.

Proof. The result follows from (3.19) immediately.

4. Stability of Biwave Maps

Let be the coordinates of a point in a compact space-time domain and let be the frame at the point. Suppose that is a biwave map from a compact space-time domain into a Riemannian manifold such that the compact supports of and are contained in the interior of Let be a vector field such that . If we apply the second variation of a biharmonic map in [4] to a biwave map, we can have the following.

Lemma 4.1. If is a biwave map from a compact domain into a Riemannian manifold, then where is the Riemannian connection on and is the vector field along

Definition 4.2. Let be a biwave map. If then is a stable biwave map.
If we consider a wave map, that is, as a biwave map, then by (4.1) we have and is automatically stable.

Definition 4.3. Let be a smooth map from a Minkowski space into a Riemannian manifold The stress energy is defined by where is the energy function and The map satisfies the conservation โ€‰โ€‰law if

Proposition 4.4. Let be a smooth map from a Minkowski space into a Riemannian manifold Then

Proof. Let be the coordinates of a point in and where is an matrix. We calculate where the first term and the third term are canceled out and

Theorem 4.5. Let be a compact domain and let be a Riemannian manifold with constant sectional curvature If is a stable biwave map satisfying the conservation law, then is a wave map.

Proof. Because has constant sectional curvature, the second term of (4.1) disappears and (4.1) becomes In particular, let Recalling that is a biwave map and has constant sectional curvature , (4.4) can be reduced to Since satisfies the conservation law, by Definition 4.3, Proposition 4.4, and (4.2) we have Substituting (4.6) into (4.5) and applying the stability of , we get which implies that that is, is a wave map.

If we apply the Hessian of the bi-energy of a biharmonic map [4] to a biwave map , then we have the following.

Lemma 4.6. Let be a biwave map. The Hessian of the bi-energy functional of is where for

Theorem 4.7. Let be a wave map on a compact domain with constant energy and let be an inclusion map. Then is an unstable biwave map.

Proof. We have the following identities from Theorem 3.5: Then we obtain the following formula from Lemma 4.6 and the previous identities: which is strictly negative, where . Hence, is an unstable biwave map.

Acknowledgment

The author would like to appreciate Professor Jie Xiao and the referees for their comments.

References

  1. J. Eells, Jr. and J. H. Sampson, โ€œHarmonic mappings of Riemannian manifolds,โ€ American Journal of Mathematics, vol. 86, no. 1, pp. 109โ€“160, 1964. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. J. Eells and L. Lemaire, โ€œA report on harmonic maps,โ€ The Bulletin of the London Mathematical Society, vol. 10, no. 1, pp. 1โ€“68, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. J. Eells and L. Lemaire, โ€œAnother report on harmonic maps,โ€ The Bulletin of the London Mathematical Society, vol. 20, no. 5, pp. 385โ€“524, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. G. Y. Jiang, โ€œ2-harmonic maps and their first and second variational formulas,โ€ Chinese Annals of Mathematics. Series A, vol. 7, no. 4, pp. 389โ€“402, 1986. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. G. Y. Jiang, โ€œ2-harmonic isometric immersions between Riemannian manifolds,โ€ Chinese Annals of Mathematics. Series A, vol. 7, no. 2, pp. 130โ€“144, 1986. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. R. Caddeo, S. Montaldo, and C. Oniciuc, โ€œBiharmonic submanifolds of 𝕊3,โ€ International Journal of Mathematics, vol. 12, no. 8, pp. 867โ€“876, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. R. Caddeo, C. Oniciuc, and P. Piu, โ€œExplicit formulas for non-geodesic biharmonic curves of the Heisenberg group,โ€ Rendiconti del Seminario Matematico. Università e Politecnico di Torino, vol. 62, no. 3, pp. 265โ€“277, 2004. View at: Google Scholar | MathSciNet
  8. E. Loubeau and C. Oniciuc, โ€œOn the biharmonic and harmonic indices of the Hopf map,โ€ Transactions of the American Mathematical Society, vol. 359, no. 11, pp. 5239โ€“5256, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. S. Montaldo and C. Oniciuc, โ€œA short survey on biharmonic maps between Riemannian manifolds,โ€ Revista de la Unión Matemática Argentina, vol. 47, no. 2, pp. 1โ€“22, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. Y.-J. Chiang and R. Wolak, โ€œTransversally biharmonic maps between foliated Riemannian manifolds,โ€ International Journal of Mathematics, vol. 19, no. 8, pp. 981โ€“996, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  11. Y.-J. Chiang and H. Sun, โ€œ2-harmonic totally real submanifolds in a complex projective space,โ€ Bulletin of the Institute of Mathematics. Academia Sinica, vol. 27, no. 2, pp. 99โ€“107, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. Y.-J. Chiang and H. Sun, โ€œBiharmonic maps on V-manifolds,โ€ International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 8, pp. 477โ€“484, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. S.-Y. A. Chang, L. Wang, and P. C. Yang, โ€œA regularity theory of biharmonic maps,โ€ Communications on Pure and Applied Mathematics, vol. 52, no. 9, pp. 1113โ€“1137, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. C. Wang, โ€œBiharmonic maps from R4 into a Riemannian manifold,โ€ Mathematische Zeitschrift, vol. 247, no. 1, pp. 65โ€“87, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. C. Wang, โ€œStationary biharmonic maps from Rm into a Riemannian manifold,โ€ Communications on Pure and Applied Mathematics, vol. 57, no. 4, pp. 419โ€“444, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. S. Klainerman and M. Machedon, โ€œSmoothing estimates for null forms and applications,โ€ Duke Mathematical Journal, vol. 81, no. 1, pp. 99โ€“133, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. S. Klainerman and M. Machedon, โ€œOn the optimal local regularity for gauge field theories,โ€ Differential and Integral Equations, vol. 10, no. 6, pp. 1019โ€“1030, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  18. J. Shatah and M. Struwe, Geometric Wave Equations, Courant Lecture Notes in Mathematics, 2, Courant Institute of Mathematical Sciences, New York, NY, USA, 2000. View at: Zentralblatt MATH
  19. J. Shatah and M. Struwe, โ€œThe Cauchy problem for wave maps,โ€ International Mathematics Research Notices, vol. 2002, no. 11, pp. 555โ€“571, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. T. Tao, โ€œGlobal regularity of wave maps. I. Small critical Sobolev norm in high dimension,โ€ International Mathematics Research Notices, no. 6, pp. 299โ€“328, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  21. T. Tao, โ€œGlobal regularity of wave maps. II. Small energy in two dimensions,โ€ Communications in Mathematical Physics, vol. 224, no. 2, pp. 443โ€“544, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. D. Tataru, โ€œThe wave maps equation,โ€ Bulletin of the American Mathematical Society, vol. 41, no. 2, pp. 185โ€“204, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  23. D. Tataru, โ€œRough solutions for the wave maps equation,โ€ American Journal of Mathematics, vol. 127, no. 2, pp. 293โ€“377, 2005. View at: Publisher Site | Google Scholar | MathSciNet
  24. A. Nahmod, A. Stefanov, and K. Uhlenbeck, โ€œOn the well-posedness of the wave map problem in high dimensions,โ€ Communications in Analysis and Geometry, vol. 11, no. 1, pp. 49โ€“83, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  25. Y.-J. Chiang and Y.-H. Yang, โ€œExponential wave maps,โ€ Journal of Geometry and Physics, vol. 57, no. 12, pp. 2521โ€“2532, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  26. Y.-J. Chiang and R. Wolak, โ€œTransversal wave maps,โ€ preprint. View at: Google Scholar
  27. T. Ichiyama, J.-I. Inoguchi, and H. Urakawa, โ€œClassification and isolation phenomena of biharmonic maps and bi-Yang-Mills fields,โ€ preprint. View at: Google Scholar
  28. Y. L. Xin and X. P. Chen, โ€œThe hypersurfaces in the Euclidean sphere with relative affine Gauss maps,โ€ Acta Mathematica Sinica, vol. 28, no. 1, pp. 131โ€“139, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  29. L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998. View at: Zentralblatt MATH | MathSciNet
  30. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1983. View at: Zentralblatt MATH | MathSciNet

Copyright © 2009 Yuan-Jen Chiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

577ย Views | 420ย Downloads | 1ย Citation
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.