Abstract
Let M = (M m, g), N = (N n, h) be C β Riemannian manifolds of dimensions m,n respectively and let ΓΈ: M m β N n be a C β mapping between them.
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Bibliography
A. Bahy-El-Dien and J.C. Wood, The explicit construction of all harmonic 2-spheres in G2(βn), J. Reine u. Angew. Math. 398 (1989), 36β66.
A. Bahy-El-Dien and J.C. Wood, The explicit construction of all harmonic two-spheres in quaternionic projective space, Proc. London Math. Soc. 62 (1991), 202β224.
P. Baird, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Research Notes in Math. 87, Pitman, London (1983).
P. Baird and J. Eells, A conservation law for harmonic maps,in: Geometry Symposium, Utrecht 1980, ed. E. Looijenga, D. Siersma and F. Takens, Lecture Notes in Math. 894, Springer, (1981), 1β25.
P. Baird and A. Ratto, Conservation laws, equivariant harmonic maps and harmonic morphisms, Proc. London Math. Soc. (3) 64 (1992), 197β224.
P. Baird and J.C. Wood, Bernstein theorems for harmonic morphisms from II83 and S3, Math. Ann. 280 (1988), 579β603.
P. Baird and J.C. Wood, Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms, J. Austral. Math. Soc. (A) 51 (1991), 118β153.
P. Baird and J.C. Wood, Harmonic morphisms, Seifert fibre spaces and conformal foliations, Proc. London Math. Soc. (3) 64 (1992), 170β196.
P. Baird and J.C. Wood, The geometry of a pair of Riemannian foliations by geodesics and asociated harmonic morphisms, Bull. Soc. Math. Belg. Ser. B 44 (1992), 115β139.
A.I. Bobenko, All constant mean curvature tori in R 3, S 3, H 3 in terms of theta-functions, Math. Ann. 290 (1991), 209β245.
J. Bolton, F. Pedit and L.M. Woodward, Minimal surfaces and the Toda field model,preprint, Universities of Durham and Massachussetts, Amherst.
J. Bolton and L.M. Woodward, The affine Toda equations and minimal surfaces,this volume.
M. Bordemann, M. Forger, J. Laartz, U. SchΓ€per, 2-dimensional nonlinear sigma models: zero curvature and Poisson structure,this volume.
R. Bryant, Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Diff. Geom 17 (1982), 455β473.
F.E. Burstall, Twistor methods for harmonic maps, in: Differential Geometry (Proceedings, Lyngby 1985), ed. V.L. Hansen, Lecture Notes in Math. 1263, Springer, Berlin (1987), 55β96.
F.E. Burstall, Harmonic maps and soliton theory, Mathematica Contemporanea 2 (1992), 1β18.
F.E. Burstall, Harmonic tori in spheres and complex projective spaces,in preparation.
F.E. Burstall, D. Ferus, F. Pedit and U. Pinkall, Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras, Ann. of Math. 138 (1993), 173β212.
F.E. Burstall and F. Pedit, Harmonic maps via Adler-Kostant-Symes theory,this volume.
F.E. Burstall and J.H. Rawnsley, Twistor theory for Riemannian symmetric spaces with applications to harmonic maps of Riemann surfaces, Lecture Notes in Math. 1424, Springer, Berlin 1990.
F.E. Burstall and S.M Salamon, Tournaments, flags and harmonic maps, Math. Ann. 277 (1987), 249β266.
F. Burstall and J.C. Wood, The construction of harmonic maps into complex Grassmannians, J. Diff. Geom. 23 (1986), 255β298.
E. Calabi, Quelques applications de lβanalyse complexe aux surfaces dβaire minima in: Topics in Complex Manifolds, UniversitΓ© de MontrΓ©al (1967), 59β81.
J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam 1975.
Y-M. Chen and W-Y. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990), 567β578.
S.S. Chern, Minimal surfaces in Euclidean space of N dimensions, in: Symposium in honor of Marston Morse, Princeton University Press 1965, 187β198.
S.S. Chern and J. Wolfson, Harmonic maps of the two-sphere into a complex Grassmannian manifold II, Ann. of Math. 125 (1987), 301β335.
J.-M. Coron and J.-M. Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques, C.R. Acad. Sci. Paris SΓ©rie 1, Math. 308 (1989), 339β344.
A. Doliwa and A. Sym, The non-linear o - -model in spheres, preprint, Warsaw University, 1992.
B.A. Dubrovin, Theta functions and non-linear equations, Uspekhi Mat. Nauk 36: 2 (1981), 11β80
B.A. Dubrovin, Theta functions and non-linear equations, Russian Math, Surveys 36: 2 (1981), 11β92.
J. Fells, On the surfaces of Delaunay and their Gauss maps, Proc. 4th Inter. Colloq. Diff. Geom., Santiago de Compostela (1978), Univ. de Santiago de Compostela 1979, 97β116; reprinted as The surfaces of Delauney, Math. Intell. 9 (1978), 53β57.
J. Fells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1β68.
J. Eells and L. Lemaire, Selected topics in harmonic maps, C.B.M.S. Regional Conference Series 50, Amer. Math. Soc. 1983.
J. Fells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385β524.
J. Fells and A. Ratto, Harmonic maps and minimal immersions with symmetries, Methods of ordinary diferential equations applied to elliptic variational problems, Annals of Math. Studies, 130, Princeton Univ. Press 1993.
J. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109β160.
J. Eells and J.C. Wood, Restrictions on harmonic maps of surfaces, Topology 15 (1976), 263β266.
J. Eells and J.C. Wood, Harmonic maps from surfaces to complex projective spaces, Adv. in Math. 49 (1983), 217β263.
G.F.R. Ellis and S.W. Hawking, The large scale structure of space time, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press (1973).
S. Erdem and J.C. Wood, On the construction of harmonic maps into a Grassmannian,J. London Math. Soc. (2) 28 (1983), 161β174.
J. Eschenburg and R. Tribuzy, Branch points of conformal mappings of surfaces, Math. Ann. 278 (1988), 621β633.
D. Ferus, F. Pedit, U. Pinkall and I. Sterling, Minimal tori in S 4, J. Reine Angew. Math. 429 (1992), 1β47.
A.P. Fordy (ed.) Soliton theory: a survey of results, Manchester Univ. Press 1990.
P.A. Griffiths, Linearizing flows and a cohomological interpretation of Lax equations,Amer. J. Math. 107 (1985), 1445β1483.
J.F. Growtowski, Harmonic map heat flow for axially symmetric data, Man. Math. 73 (1991), 207β228.
C.-H. Gu, On the harmonic maps from Rβ to S 1 β, J. Reine u. Angew. Math. 346 (1984), 101β109.
M. Guest, Harmonic 2-spheres in complex projective space and some open problems, Expos. Math. 10 (1992), 61β87.
R. Gulliver, R. Osserman and H.L. Royden, A theory of branched immersions of surfaces, Amer. J. Math. 95 (1973), 750β812.
F. HΓ©lein, RΓ©gularitΓ© des applications faiblement harmoniques entre une surface et une variΓ©tΓ© riemannienne, C.R. Acad. Sci. Paris 312 (1991), 591β596.
D. Hilbert, Die Grundlagen der Physik, Nachr. Ges. Wiss. GΓΆttingen (1915), 395β407, (1917), 53β76.
N.J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Diff. Geom. 31 (1990), 627β710.
D.A. Hoffman and R. Osserman, The Gauss map of surfaces in R 3 and R4, Proc. London Math. Soc. 50 (1985), 27β57.
W-Y. Hsiang and H.B. Lawson, Minimal submanifolds of low cohomogeneity, J. Diff. Geom. 5 (1971), 1β38.
H. Karcher and J.C. Wood, Non-existence results and growth properties for harmonic maps and forms, J. Reine u. Angew. Math. 353 (1984), 165β180.
K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), 89β99.
P.Z. Kobak, Twistors, nilpotent orbits and harmonic maps, this volume.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry,vol II, Inter-science, Wiley, New York 1969.
A.O. Kowalski, Generalized symmetric spaces, Lecture Notes in Math. 805, Springer-Verlag, Berlin (1980).
B. Loo, The space of harmonic maps of S 2 into S 4, Trans. Amer. Math. Soc. 313 (1989), 81β102.
A. Lichnerowicz, Applications harmoniques et variΓ©tΓ©s KΓ€hlΓ©riennes, Symp. Math. III (Bologna 1970 ), 341β402.
M. Mafias, The principal chiral model as an integrable system,this volume.
H. McKean and E. Trubowitz, Hillβs equation and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143β226.
I. McIntosh, Infinite dimensional Lie groups and the two-dimensional Toda lattice,this volume.
M. Melko and I. Sterling, Integrable systems and the classical theory of surfaces,this volume.
J. Moser, Various aspects of integrable Hamiltonian systems, Progress in Math. 8, BirkhΓ€user, Boston 1980.
Y. Ohnita and G. Valli, Pluriharmonic maps into compact Lie groups and factorization into unitons, Proc. London Math. Soc. (3) 61 (1990), 546β570.
K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207β221.
J.H. Rawnsley, f-structures, f-twistor spaces and harmonic maps, Sem. Geom. L. Bianchi II, 1984, Lecture Notes in Math. 1164, Springer, Berlin 1985, 85β159.
J.H. Rawnsley, Noetherβs theorem for harmonic maps, in: Diff. Geom. Methods in Math. Phys., ed. S. Sternberg, Reidel, Dodrecht 1984, 197β202.
A.G. Reyman and M.A. Semenov-Tian-Shansky, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, II, Invent. Math. 63 (1981), 423β432.
C. Rogers and W.F. Shadwick, BΓ€cklund transforms and their applications, Academic Press, New York 1982.
E.A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569β573.
R.T. Smith, Harmonic maps of spheres, Thesis, University of Warwick, 1972.
I. Sterling and H.C. Wente, Existence and classification of constant mean curvature multibubbletons of finite and infinite type, preprint, Univ. Toledo (1992).
A.V. Tyrin, Harmonic spheres in compact Lie groups and extremals of a multivalued Novikov functional, Uspekhi Mat. Nauk. 46: 3 (1991), 197β198
A.V. Tyrin, Harmonic spheres in compact Lie groups and extremals of a multivalued Novikov functional, Russ. Math. Surveys 46: 3 (1991), 235β236.
K. Uhlenbeck, Minimal 2-spheres and tori in S e β, preprint (1975).
K. Uhlenbeck, Equivariant harmonic maps into spheres, Harmonic maps, Proceedings, New Orleans 1980, ed. R.J. Knill, M. Kalka and H.C.J. Sealey, Lecture Notes in Math. 949, Springer, Berlin 1982, 146β158.
K. Uhlenbeck, Harmonic maps into Lie groups (classical solutions of the chiral model), J. Diff. Geom. 30 (1989), 1β50.
M. Umehara and K. Yamada, Harmonic non-holomorphic mappings of 2-tori into the 2-sphere, in: Geometry of Manifolds (Matsumoto 1988), ed. K. Shiohama, Perspectives in Math. 8, Academic Press, Boston, London 1989.
G. Valli, On the energy spectrum of harmonic 2-spheres in unitary groups, Topology 27 (1988), 129β136.
J. Wolfson, Harmonic maps of the two-sphere into the complex hyperquadric, J. Diff. Geom. 24 (1986), 141β152.
J. Wolfson, Harmonic sequences and harmonic maps of surfaces into complex Grassmannian manifolds, J. Diff. Geom. 27 (1988), 161β178.
J.C. Wood, Twistor constructions for harmonic maps, in: Differential Geometry and Differential Equations (Proceedings, Shanghai, 1985), ed. Gu Chaohao, M. Berger and R.L. Bryant, Lecture Notes in Math. 1255, Springer, Berlin (1987), 130β159.
J.C. Wood, Holomorphic differentials and classification theorems for harmonic maps and minimal immersions, in: Global Riemannian Geometry, ed. T.J. Willmore and N.J. Hitchin, Ellis Horwood, Chichester 1984, 168β175.
J.C. Wood, The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmannian, J. Reine u. Angew. Math. 386 (1988), 1β31.
J.C. Wood, Explicit construction and parametrization of harmonic 2-spheres in the unitary group, Proc. London Math. Soc. (3) 58 (1989), 608β624.
J.C. Wood, Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Internat. J. Math. 3 (1992), 415β439.
J.C. Wood, Harmonic maps and harmonic morphisms,Lobachevshy Semester, Euler International Math. Inst. (to appear).
V.E. Zakharov and A.V. Mikhailov, Relativistically invariant two-dimensional model of field theory which is integrable by means of the inverse scattering problem method, Zh. Eksp. Teor.Fiz. 74 (1978), 1953β1973
V.E. Zakharov and A.V. Mikhailov, Relativistically invariant two-dimensional model of field theory which is integrable by means of the inverse scattering problem method, Sov. Phys. JETP 47 (1978) 1017β1027.
V.E. Zakharov and A.V. Shabat Integration of nonlinear equations of MathematicalβPhysics by inverse scattering II, Funkts. Anal. Prilozh. 13 (1978), 13β22
V.E. Zakharov and A.V. Shabat Integration of nonlinear equations of MathematicalβPhysics by inverse scattering II, Func. Anal. Appl. 13 (1979), 166β174.
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Wood, J.C. (1994). Harmonic maps into symmetric spaces and integrable systems. In: Fordy, A.P., Wood, J.C. (eds) Harmonic Maps and Integrable Systems. Aspects of Mathematics, vol E 23. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14092-4_3
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DOI: https://doi.org/10.1007/978-3-663-14092-4_3
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