Analytic Aspects of the Harmonic Map Problem Conference paper DOI :
10.1007/978-1-4612-1110-5_17

Part of the
Mathematical Sciences Research Institute Publications
book series (MSRI, volume 2) Cite this paper as: Schoen R.M. (1984) Analytic Aspects of the Harmonic Map Problem. In: Chern S.S. (eds) Seminar on Nonlinear Partial Differential Equations. Mathematical Sciences Research Institute Publications, vol 2. Springer, New York, NY Abstract A fundamental nonlinear object in differential geometry is a map between manifolds. If the manifolds have Riemannian metrics, then it is natural to choose representaives for maps which respect the metric structures of the manifolds. Experience suggests that one should choose maps which are minima or critical points of variational integrals. Of the integrals which have been proposed, the energy has attracted most interest among analysts, geometers, and mathematical physicists. Its critical points, the harmonic maps, are of some geometric interest. They have also proved to be useful in applications to differential geometry. Particularly one should mention the important role they play in the classical minimal surface theory. Secondly, the applications to Kahler geometry given in [S], [SiY] illustrate the usefulness of harmonic maps as analytic tools in geometry. It seems to the author that there is good reason to be optimistic about the role which the techniques and results related to this problem can play in future developments in geometry.

References [BG]

H. J. Borchers and W. J. Garber, Analyticity of solutions of the O(N) non-linear σ-model, Comm. Math. Phys. 71(1980), 299–309.

MathSciNet MATH CrossRef Google Scholar [CG]

S. S. Chern and S. Goldberg, On the volume-decreasing property of a class of real harmonic mappings, Amer. J. Math. 97(1975), 133–147.

MathSciNet MATH CrossRef Google Scholar [ES]

J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86(1964), 109–160.

MathSciNet MATH CrossRef Google Scholar [EL]

J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10(1978), 1–68.

MathSciNet MATH CrossRef Google Scholar [F]

J. Frehse, A discontinuous solution to a mildly nonlinear elliptic system, Math. Z. 134(1973), 229–230.

MathSciNet MATH CrossRef Google Scholar [G]

M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies 105, 1983.

[GG1]

M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta. Math. 148(1982), 31–46.

MathSciNet MATH CrossRef Google Scholar [GG2]

M. Giaquinta and E. Giusti, The singular set of the minima of certain quadratic functionals, to appear in Ann. Sc. Norm. Sup. Pisa.

[GH]

M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 336(1982), 124–164.

MathSciNet Google Scholar [Hm]

R. Hamilton, Harmonic maps of manifolds with boundary, Lecture notes 471, Springer 1975.

[Hr]

P. Hartman, On homotopic harmonic maps, Can. J. Math. 19(1967), 673–687.

MathSciNet MATH CrossRef Google Scholar [HKW]

S. Hildebrandt, H. Kaul, and K. O. Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138(1977), 1–16.

MathSciNet MATH CrossRef Google Scholar [HW]

S. Hildebrandt and K. O. Widman, On the Holder continuity of weak solutions of quasilinear elliptic systems of second order, Ann. Sc. Norm. Sup. Pisa IV(1977), 145–178.

MathSciNet Google Scholar [JM]

J. Jost and M. Meier, Boundary regularity for minima of certain quadratic functionals, Math. Ann. 262(1983), 549–561.

MathSciNet MATH CrossRef Google Scholar [L]

L. Lemaire, Applications harmoniques de surfaces Riemanniennes, J. Diff. Geom. 13(1978), 51–78.

MathSciNet MATH Google Scholar [Ml]

C. B. Morrey, On the solutions of quasilinear elliptic partial differential equations, Trans. A.M.S. 43(1938), 126–166.

MathSciNet CrossRef Google Scholar [M2]

C. B. Morrey, The problem of Plateau on a Riemannian manifold, Ann. of Math. 49(1948), 807–851.

MathSciNet MATH CrossRef Google Scholar [M3]

C. B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, New York, 1966.

MATH Google Scholar [P]

P. Price, A monotonicity formula for Yang-Mills fields, Manuscripta Math. 43(1983), 131–166.

MathSciNet MATH CrossRef Google Scholar [ScY1]

R. Schoen and S. T. Yau, Compact group actions and the topology of manifolds with non-positive curvature, Topology 18(1979), 361–380.

MathSciNet MATH CrossRef Google Scholar [ScY2]

R. Schoen and S. T. Yau, Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature, Ann. of Math. 110(1979), 127–142.

MathSciNet MATH CrossRef Google Scholar [S]

Y. T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds, Ann. of Math. 112(1980), 73–111.

MathSciNet MATH CrossRef Google Scholar [SiY]

Y. T. Siu and S. T. Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59(1980), 189–204.

MathSciNet MATH CrossRef Google Scholar [SaU]

J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113(1981), 1–24.

MathSciNet MATH CrossRef Google Scholar [SU1]

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17(1982), 307–335.

MathSciNet MATH Google Scholar [SU2]

R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18(1983), 253–268.

MathSciNet MATH Google Scholar [SU3]

R. Schoen and K. Unlenbeck, Regularity of minimizing harmonic maps into the sphere, to appear in Invent. Math.

[St]

M. Struwe, preprint.

[W]

B. White, Homotopy classes in Sobolev spaces of mappings, preprint.

© Springer Science+Business Media New York 1984

Authors and Affiliations 1. Mathematics Department University of California Berkeley USA