Group actions and deformations for harmonic maps
Martin A. GUEST, Yoshihiro OHNITA
著者情報
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Martin A. GUEST
Department of Mathematics University of Rochester
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Yoshihiro OHNITA
Department of Mathematics Tokyo Metropolitan University
ジャーナル
フリー
1993 年 45 巻 4 号 p. 671-704
詳細
- Published: 1993 Received: 1992/03/12 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
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訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報 訂正内容: Wrong : [AJS] G. Arsenault, M. Jacques and Y. Saint-Aubin, Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian sigma models, J. Math. Phys., 29 (1988), 1465-1471.
[AP] M. F. Atiyah and A. N. Pressley, Convexity and loop groups, Arithmetic and Geometry, Vol. 11, (eds. M. Artin and J. Tate), Progr. Math. 36, Birkhauser, 1983, pp. 33-64.
[AS1] G. Arsenault and Y. Saint-Aubin, The hidden symmetry of U(n) principal σ models revisited: I. Explicit expressions for the generators, Nonlinearity, 2 (1989), 571-591.
[AS2] G. Arsenault and Y. Saint-Aubin, The hidden symmetry of U(n) principal σ models revisited: 11. The algebraic structure, Nonlinearity, 2 (1989), 593-607.
[Ba] J. L. M. Barbosa, On minimal immersions of S2 into S2m, Trans. Amer. Math. Soc., 210 (1975), 75-106.
[Be] E. J. Beggs, Solitons in the chiral equation, Comm. Math. Phys., 128 (1990), 131-139.
[BG] M. J. Bergvelt and M. A. Guest, Actions of loop groups on harmonic maps, Trans. Amer. Math. Soc., 326 (1991), 861-886.
[Bo] R. Bott, An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France, 84 (1956;), 251-281.
[Br] R. L. Bryant, Lie groups and twistor spaces, Duke Math. J., 52 (1985), 223-261.
[BR] F. E. Burstall and J. H. Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Math., 1424, Springer, 1990.
[Ca1] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom., 1 (1967), 111-125.
[Ca2] E. Calabi, Quelques applications de l'analyse complexe aux surfaces d'aire minima, Topics in Complex Manifolds, Université de Montréal, 1968, pp. 59-81.
[EK] N. Ejiri and M. Kotani, Minimal surfaces in S2m(1) with extra eigenfunctions, preprint, Nagoya University and Toho University.
[EL] J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524.
[FL] W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math., 146 (1981), 271-283.
[Fr] T. Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math., 70 (1959), 1-8.
[Gu] M. A. Guest, Geometry of snaps between generalized flag manifolds, J. Differential Geom., 25 (1987), 223-247.
[JK] M. Jacques and Y. Saint-Aubin, Infinite-dimensional Lie algebras acting on the solution space of various σ models, J. Math. Phys., 28 (1987), 2463-2479.
[Ko] R. R. Kocherlakota, Integral homology of real flag manifolds and loop spaces of symmetric spaces, preprint, Harvard University.
[Kt] M. Kotani, Connectedness of the space of minimal 2-spheres in S2m(1), to appear, Proc. Amer. Math. Soc..
[Lo] B. Loo, The space of harmonic maps of S2 into S4, Trans. Amer. Math. Soc., 313 (1989), 81-102.
[Mi] S. A. Mitchell, A filtration of the loops on SU(N) by Schubert varieties, Math. Z., 193 (1986), 347-362.
[Na] M. Namba, Families of Meromorphic Functions on Compact Riemann Surfaces, Lecture Notes in Math., 767, Springer, 1979.
[OV] Y. Ohnita and G. Valli, Pluriharmonic maps into compact Lie groups and factorization into unitons, Proc. London Math. Soc., 61 (1990), 546-570.
[PS] A. N. Pressley and G. B. Segal, Loop Groups, Oxford University Press, 1986.
[Ri] W. Richter, Attaching maps for the disk bundles of some Morse-Bott decompositions, Ph. D, thesis, University of Washington, 1988.
[Se] G. B. Segal, Loop groups and harmonic maps, Advances in Homotopy Theory, L. M. S. Lecture Notes, 139, Cambridge Univ. Press, 1989, pp. 153-164.
[SU] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math., 113 (1981), 1-24.
[To] G. W. Towsley, Conformal deformations of meromorphic functions, Ph. D, thesis, University of Rochester, 1975.
[Uh] K. Uhlenbeck, Harmonic maps into Lie groups (Classical solutions of the chiral model), J. Differential Geom., 30 (1989), 1-50.
[Va] G. Valli, On the energy spectrum of harmonic 2-spheres in unitary groups, Topology, 27 (1988), 129-136.
[Ve1] J. L. Verdier, Two dimensional σ-models and harmonic maps from S2 to S2n, Lecture Notes in Phys., 180, Springer, 1983, pp. 136-141.
[Ve2] J. L. Verdier, Applications harmoniques de S2 dans S4, Geometry of Today, Giornate di Geometria, Roma 1984, (eds. E. Arbarello, C. Procesi and E. Strickland), Birkhäuser, 1985, pp. 267-282.
[Ve3] J. L. Verdier, Applications harmoniques de S2 dans S4: II, Harmonic Mappings, Twistors, and σ-models, Advanced Series in Math. Phys., 4, (ed. P. Gauduchon), World Scientific, Singapore, 1988, pp. 124-147.
[ZM] V. E. Zakharov and A. V. Mikhailov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP, 47 (1978), 1017-1027.
[ZS] V. E. Zakharov and A. B. Shabat, Integration of non-linear equations of mathematical physics by the inverse scattering method II, Funct. Anal. Appl., 13 (1979), 13-22.
Right : [AJS] G. Arsenault, M. Jacques and Y. Saint-Aubin, Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian sigma models, J. Math. Phys., 29 (1988), 1465-1471.
[AP] M. F. Atiyah and A. N. Pressley, Convexity and loop groups, Arithmetic and Geometry, Vol. II, (eds. M. Artin and J. Tate), Progr. Math. 36, Birkhäuser, 1983, pp. 33-64.
[AS1] G. Arsenault and Y. Saint-Aubin, The hidden symmetry of U(n) principal σ models revisited: I. Explicit expressions for the generators, Nonlinearity, 2 (1989), 571-591.
[AS2] G. Arsenault and Y. Saint-Aubin, The hidden symmetry of U(n) principal σ models revisited: II. The algebraic structure, Nonlinearity, 2 (1989), 593-607.
[Ba] J. L. M. Barbosa, On minimal immersions of S2 into S2m, Trans. Amer. Math. Soc., 210 (1975), 75-106.
[Be] E. J. Beggs, Solitons in the chiral equation, Comm. Math. Phys., 128 (1990), 131-139.
[BG] M. J. Bergvelt and M. A. Guest, Actions of loop groups on harmonic maps, Trans. Amer. Math. Soc., 326 (1991), 861-886.
[Bo] R. Bott, An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France, 84 (1956), 251-281.
[Br] R. L. Bryant, Lie groups and twistor spaces, Duke Math. J., 52 (1985), 223-261.
[BR] F. E. Burstall and J. H. Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Math., 1424, Springer, 1990.
[Ca1] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom., 1 (1967), 111-125.
[Ca2] E. Calabi, Quelques applications de l'analyse complexe aux surfaces d'aire minima, Topics in Complex Manifolds, Université de Montréal, 1968, pp. 59-81.
[EK] N. Ejiri and M. Kotani, Minimal surfaces in S2m(1) with extra eigenfunctions, preprint, Nagoya University and Toho University.
[EL] J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524.
[FL] W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math., 146 (1981), 271-283.
[Fr] T. Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math., 70 (1959), 1-8.
[Gu] M. A. Guest, Geometry of maps between generalized flag manifolds, J. Differential Geom., 25 (1987), 223-247.
[JK] M. Jacques and Y. Saint-Aubin, Infinite-dimensional Lie algebras acting on the solution space of various σ models, J. Math. Phys., 28 (1987), 2463-2479.
[Ko] R. R. Kocherlakota, Integral homology of real flag manifolds and loop spaces of symmetric spaces, preprint, Harvard University.
[Kt] M. Kotani, Connectedness of the space of minimal 2-spheres in S2m(1), to appear, Proc. Amer. Math. Soc..
[Lo] B. Loo, The space of harmonic maps of S2 into S4, Trans. Amer. Math. Soc., 313 (1989), 81-102.
[Mi] S. A. Mitchell, A filtration of the loops on SU(N) by Schubert varieties, Math. Z., 193 (1986), 347-362.
[Na] M. Namba, Families of Meromorphic Functions on Compact Riemann Surfaces, Lecture Notes in Math., 767, Springer, 1979.
[OV] Y. Ohnita and G. Valli, Pluriharmonic maps into compact Lie groups and factorization into unitons, Proc. London Math. Soc., 61 (1990), 546-570.
[PS] A. N. Pressley and G. B. Segal, Loop Groups, Oxford University Press, 1986.
[Ri] W. Richter, Attaching maps for the disk bundles of some Morse-Bott decompositions, Ph. D. thesis, University of Washington, 1988.
[Se] G. B. Segal, Loop groups and harmonic maps, Advances in Homotopy Theory, L. M. S. Lecture Notes, 139, Cambridge Univ. Press, 1989, pp. 153-164.
[SU] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math., 113 (1981), 1-24.
[To] G. W. Towsley, Conformal deformations of meromorphic functions, Ph. D. thesis, University of Rochester, 1975.
[Uh] K. Uhlenbeck, Harmonic maps into Lie groups (Classical solutions of the chiral model), J. Differential Geom., 30 (1989), 1-50.
[Va] G. Valli, On the energy spectrum of harmonic 2-spheres in unitary groups, Topology, 27 (1988), 129-136.
[Ve1] J. L. Verdier, Two dimensional σ-models and harmonic maps from S2 to S2n, Lecture Notes in Phys., 180, Springer, 1983, pp. 136-141.
[Ve2] J. L. Verdier, Applications harmoniques de S2 dans S4, Geometry of Today, Giornate di Geometria, Roma 1984, (eds. E. Arbarello, C. Procesi and E. Strickland), Birkhäuser, 1985, pp. 267-282.
[Ve3] J. L. Verdier, Applications harmoniques de S2 dans S4: II, Harmonic Mappings, Twistors, and σ-models, Advanced Series in Math. Phys., 4, (ed. P. Gauduchon), World Scientific, Singapore, 1988, pp. 124-147.
[ZM] V. E. Zakharov and A. V. Mikhailov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP, 47 (1978), 1017-1027.
[ZS] V. E. Zakharov and A. B. Shabat, Integration of non-linear equations of mathematical physics by the inverse scattering method II, Funct. Anal. Appl., 13 (1979), 13-22.
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