Biharmonic Maps
 YuanJen Chiang
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Abstract
The last five decades have witnessed many developments in the theory of harmonic maps. To become acquainted to some of these, the reader is referred to two reports and a survey paper by Eells and Lemaire [119, 122, 124] about the developments of harmonic maps up to 1988 for details. Several books on harmonic maps [203, 205, 206, 389, 425] are also available. In this chapter, we follow the notions and notations of harmonic maps between Riemannian manifolds by Eells Sampson [129] in the introduction.We discuss the crucial topics in harmonic maps including fundamentals, regularity, maps of surfaces, maps of KRahler manifolds, maps into groups and Grassmannians, harmonic maps, loop groups, and integrable systems, harmonicmorphisms, maps of singular spaces, and transversally harmonic maps. Since the theory of harmonic maps has been developed over half a century, it is impossible to provide full details. However, we try to present the most important components of the topics.
 K. Arslan, R. Ezentas, C. Murathan, T. Sasahara, Biharmonic submanifolds in 3dimensional (k, μ)manifolds. Int. J. Math. Math. Sci. 22, 3575–3586 (2005) CrossRef
 K. Arslan, R. Ezentas, C. Murathan, T. Sasahara, Biharmonic antiinvariant submanifolds in Sasakian space forms. Beitr. Algebra Geom. 48(1), 191–207 (2007)
 P. Baird, J. Eells, A Conservation Law for Harmonic Maps. Lecture Notes in Mathematics, vol. 894 (Springer, Berlin, 1981), pp. 1–25
 P. Baird, D. Kamissoko, On constructing biharmonic maps and metrics. Ann. Glob. Anal. Geom. 23(1), 65–75 (2003) CrossRef
 A. Balmuç, Biharmonic properties and conformal changes. An. Stiint. Univ. Al. I. Cuza Iaçi Mat. (N.S.) 50(2), 361–272 (2004)
 A. Balmuç, S. Montaldo, C. Oniciuc, Biharmonic maps between warped product manifolds. J. Geom. Phys. 57(2), 449–466 (2007) CrossRef
 A. Balmusç, On the biharmonic curves of the Euclidean and Berger 3dimensional spheres. Sci. Ann. Univ. Agric. Sci. Vet. Med. 47(1), 87–96 (2004)
 J. Berndt, Real hypersurfaces in quaternionic space space forms. J. Reine Angew. Math. 419, 9–26 (1991)
 J. Berndt, F. Tricerri, L. Vanhecke, Generalized Heisenberg Groups and DamekRicci Harmonic Spaces. Lecture Notes in Mathematics (Springer, Berlin, 1598)
 H. Brezis, J.M. Coron, E.H. Lieb, Estimation d’énergie pour les applications de R ^{3} a valeurs dans S ^{2}. C. R. Acad. Sci. Paris 303(5), 207–210 (1986)
 R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds of S ^{3}. Int. J. Math. 12(8), 867–876 (2001) CrossRef
 R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds in spheres. Isr. J. Math. 130, 109–123 (2002) CrossRef
 R. Caddeo, C. Oniciuc, P. Piu, Explicit formulas for nongeodesic biharmonic curves of the Heisenberg group. Rend. Sem. Mat. Univ. Politec. Torino 62(3), 265–278 (2004)
 S.Y.A. Chang, L. Wang, P.C. Yang, Regularity of harmonic maps. Commun. Pure Appl. Math. LII 52(9), 1099–1111 (1999)
 S.Y.A. Chang, L. Wang, P.C. Yang, Regularity of biharmonic maps. Commun. Pure Appl. Math. LII 52(9), 1113–1137 (1999)
 B.Y. Chen, Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17(2), 169–188 (1991)
 B.Y. Chen, A report on submanifolds of finite type. Soochow J. Math. 22(2), 117–337 (1996)
 B.Y. Chen, S. Ishikawa, Biharmonic pseudoRiemannian submanifolds in pseudoEuclidean spaces. Kyushu J. Math. 52(1), 167–185 (1988) CrossRef
 Y.J. Chiang, Harmonic and biharmonic maps of Riemann surfaces. Glob. J. Pure Appl. Math. 9(2), 109–124 (2013)
 Y.J. Chiang, H.A. Sun, 2harmonic totally real submanifolds in a complex projective space. Bull. Inst. Math. Acad. Sin. 27(2), 99–107 (1999)
 Y.J. Chiang, H.A. Sun, Biharmonic maps on VManifolds. Int. J. Math. Math. Sci. 27(8), 477–484 (2001) CrossRef
 Y.J. Chiang, R. Wolak, Transversally biharmonic maps between foliated Riemannian manifolds. Int. J. Math. 19(8), 981–996 (2008) CrossRef
 J.T. Cho, J. Inoguchi, J.E. Lee, Biharmonic curves in 3dimensional Sasakian space forms. Ann. Mat. Pura. Appl. (4) 186(1), 685–700 (2007)
 Y. Dai, M. Shoji, H. Urakawa, Harmonic maps into Lie groups and homogeneous spaces. Differ. Geom. Appl. 7(2), 143–160 (1997) CrossRef
 I. Dimitric, Submanifolds of E ^{ m } with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sin. 20(1), 53–65 (1992)
 J. Eells, L. Lemaire, A report on harmonic maps. Bull. Lond. Math. Soc. 10(1), 1–68 (1978) CrossRef
 L.C. Evans, Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. 116(2), 101–113 (1991) CrossRef
 D. Fetcu, Biharmonic curves in the generalized Heisenberg group. Beitr. Algebra Geom. 46(2), 513–521 (2005)
 J. Frehse, A discontinuous solution of a mildly nonlinear elliptic system. Math. Z. 134, 229–230 (1973) CrossRef
 A. Haefliger, Pseudogroups of local isometries, differential geometry, in Proceedings of the Vth International Colloquium on Differential Geometry, Santiago de Compostela, 1984, ed. by L.A. Cordero (Pitman, Boston, 1985)
 T. Hasanis, T. Vlachos, Hypersurfaces in E ^{4} with harmonic mean curvature vector field. Math. Nachr. 72, 145–169 (1995) CrossRef
 W.Y. Hsiang, H.B. Lawson Jr., Minimal submanifolds of low cohomology. J. Differ. Geom. 5, 1–38 (1971)
 T. Ichiyama, J.I. Inoguchi, H. Urakawa, Biharmonic maps and biYangMills fields. Note Mat. 28(suppl. 1), 233–275 (2009)
 T. Ichiyama, J.I. Inoguchi, H. Urakawa, Classification and isolation phenomena of biharmonic maps and biYangMills fields. arXiv:0912.4806v1 [math.DG] 24 Dec 2009
 J.I. Inoguchi, Submanifolds with harmonic mean curvature in contact 3manifolds. Colloq. Math. 101(2), 163–179 (2004) CrossRef
 G.Y. Jiang, 2harmonic maps and their first and second variational formulas. Chin. Ann. Math. A 7(4), 389–402 (1986)
 G.Y. Jiang, 2harmonic isometric immersions between Riemannian manifolds. Chin. Ann. Math. A 7(2), 130–144 (1986)
 G.Y. Jiang, The conservation law of 2harmonic maps between Riemannian manifolds. Acta Math. Sin. 30(2), 220–225 (1987)
 F. John, L. Nirenberg, On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961) CrossRef
 S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vols. I, II (Wiley, New York, 1963/1969)
 J.J. Konderak, R. Wolak, Transversally harmonic maps between manifolds with Riemannian foliations. Q. J. Math. 54(3), 335–354 (2003) CrossRef
 Y. Ku, Interior and boundary regularity of intrinsic biharmonic maps to spheres. Pac. J. Math. 234(1), 46–67 (2008) CrossRef
 T. Lamm, Heat flow for extrinsic biharmonic maps with small initial energy. Ann. Glob. Anal. Geom. 26(4), 369–384 (2004) CrossRef
 D. Laugwitz, Differential and Riemannian Geometry (Academic, New York/London, 1965)
 A.M. Li, J.M. Li, An inequality for matrices and its applications in differential geometry. Adv. Math. (China) 20(3), 375–376 (1991)
 E. Loubau, Y.L. Ou, Biharmonic maps and morphisms from conformal mappings. Tohoku Math. J. (2) 62(1), 55–73 (2010)
 E. Loubeau, C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map. Trans. Am. Math. Soc. 359(11), 5239–5256 (2007) CrossRef
 E. Loubeau, Y.L. Ou, The characterization of biharmoinic morphisms, in Differential Geometry and Its Applications, (Opava, 2001). Math. Publ. 3 (2001), pp. 31–41
 E. Loubeau, S. Montaldo, C. Oniciuc, The stressenergy tensor for biharmonic maps. Math. Z. 259(3), 503–524 (2008) CrossRef
 S. Montaldo, C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds. Rev. Union Mat. Argent. 47(2), 1–22 (2006)
 N. Nakauchi, H. Urakawa, Removable singularities and bubbling of biharmonic maps. arXiv:0912.4086[math.DG]17 Jan 2011
 M. Obata, The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature. J. Differ. Geom. 2, 217–223 (1968)
 C. Oniciuc, Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. Al. I. Cuza Iaçi Mat. (N.S.) 48(2), 237–248 (2002)
 C. Oniciuc, On the second variation formula for biharmonic maps to a sphere. Publ. Math. Debr. 61(3–4), 613–622 (2002)
 V. Oproiu, Some classes of natural almost Hermitian structures on the tangent bundles. Publ. Math. Debr. 62(3–4), 561–576 (2003)
 Y.L. Ou, Quadratic harmonic morphisms and Osystems. Ann. Inst. Fourier (Grenoble) 47(2), 687–713 (1997)
 Y.L. Ou, pharmonic morphisms, biharmonic morphisms and nonharmonic biharmonic maps. J. Geom. Phys. 56(3), 358–374 (2006)
 Y.L. Ou, On conformal biharmonic immersions. Ann. Glob. Anal. Geom. 36(2), 133–142 (2009) CrossRef
 YL. Ou, Conformally biharmonic immersions into 3dimensional manifolds (preprint)
 Y.L. Ou, Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces. J. Geom. Phys. 62, 751–762 (2012) CrossRef
 Y.L. Ou, S. Lu, Biharmonic maps in two dimensions. Ann. Mat. doi:10.1007/s1023101102150
 Y.L. Ou, L. Tang, The generalized Chen’s conjecture on biharmonic submanifolds is false. arXiv:1006.1838v2 [math.DG] 1 Jan 2011
 Y.L. Ou, J.C. Wood, On the classification of quadrtic harmonic morphisms between Euclidean spaces. Algebras Groups Geom. 13(1), 41–53 (1996)
 J. Sacks, K. Uhlenbeck, The existence of minimal immersions of 2spheres. Ann. Math. (2) 113(1), 1–24 (1981)
 J. Sacks, K. Uhlenbeck, Minimal immersions of closed Riemann surfaces. Trans. Am. Math. Soc. 271(2), 639–652 (1982) CrossRef
 T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors. Publ. Math. Debr. 67(3–4), 285–303 (2005)
 T. Sasahara, Stability of biharmonic Legendre submanifolds in Sasakian space forms. Can. Math. Bull. 51(3), 448–459 (2008) CrossRef
 R. Schoen, Analytic aspects of the harmonic map problem, in Seminars on Nonlinear Partial Differential Equations (Berkeley, 1983). Mathematical Sciences Research Institute Publications, vol. 2 (Springer, New York, 1984), pp. 321–358
 R. Schoen, K. Uhlenbeck, A regularity theory for harmonic maps. J. Differ. Geom. 17(2), 307–335 (1982)
 Y.B. Shen, Totally real ndimensional minimal submanifold in complex ndimensional projective space. Adv. Math. (Beijing) 13(1), 65–70 (1984)
 R. Takagi, On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)
 R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures. J. Math. Soc. Jpn. 27, 43–53 (1975) CrossRef
 R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures II. J. Math. Soc. Jpn. 27(4), 507–516 (1975) CrossRef
 K. Uhlenbeck, Connections with L ^{ p }bounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982) CrossRef
 K. Uhlenbeck, Minimal spheres and other conformal variational problems, in Seminars on Minimal Submanifolds, ed. by E. Bombieri. Annals of Mathematics Studies, vol. 103 (Princeton University Press, Princeton, 1983), pp. 169–248
 H. Urakawa, Biharmonic maps into compact Lie groups and the integrable systems. arXiv:0910.0692v2 [math.DG] 5 31 Jan 2012, 1–27
 C. Wang, Biharmonic maps from R ^{4} in to a Riemannian manifold. Math. Z. 247(1), 65–87 (2004) CrossRef
 C. Wang, Stationary biharmonic maps from R ^{ m } into a Riemannian manifold. Commun. Pure Appl. Math. 57(4), 419–444 (2004) CrossRef
 Z.L. Wang, Y.L. Ou, Biharmonic Riemannian submanifolds from 3manifolds. Math. Z. 269, 917–925 (2011) CrossRef
 Y.L. Xin, X.P. Chen, The hypersurfaces in the Euclidean sphere with relative affine Gauss maps. Acta Math. Sin. 28(1), 131–139 (1985)
 Title
 Biharmonic Maps
 Book Title
 Developments of Harmonic Maps, Wave Maps and YangMills Fields into Biharmonic Maps, Biwave Maps and BiYangMills Fields
 Pages
 pp 243304
 Copyright
 2013
 DOI
 10.1007/9783034805346_4
 Print ISBN
 9783034805339
 Online ISBN
 9783034805346
 Series Title
 Frontiers in Mathematics
 Series ISSN
 16608046
 Publisher
 Springer Basel
 Copyright Holder
 Springer Basel
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 Authors

 YuanJen Chiang ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Mary Washington, Fredericksburg, VA, USA
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