On harmonic maps

The subject of harmonic maps is vast and has found many applications, and it would require a very long book to cover all aspects, even superficially. Hence, we have made a choice; in particular, highlighting the key questions of existence, uniqueness and regularity of harmonic maps between given manifolds. Thus we shall survey some of the main methods of global analysis for answering these questions. We first consider relevant aspects of harmonic functions on Euclidean space; then we give a general introduction to harmonic maps. The core of our work is in Chapters 3–6 where we present the analytical methods. We round of the article by describing how twistor theory and integrable systems can be used to construct many more harmonic maps. On the way, we mention harmonic morphisms: maps between Riemannian manifolds which preserve Laplace’s equation; these turn out to be a particular class of harmonic maps and exhibit some properties dual to those of harmonic maps. More information on harmonic maps can be found in the following articles and books; for generalities: [61, 62, 63, 219], analytical aspects: [21, 88, 103, 118, 131, 133, 135, 189, 204, 194], integrable systems methods: [73, 94, 117], applications to complex and Kähler geometry: [63, 135], harmonic morphisms: [7], and other topics: [64, 231].

[1]  Enrico Giusti,et al.  The singular set of the minima of certain quadratic functionals , 1984 .

[2]  H. Fédérer,et al.  The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension , 1970 .

[3]  Brian White,et al.  Homotopy classes in Sobolev spaces and the existence of energy minimizing maps , 1988 .

[4]  Stefan Hildebrandt,et al.  An existence theorem for harmonic mappings of Riemannian manifolds , 1977 .

[5]  Philip Hartman,et al.  On Homotopic Harmonic Maps , 1967, Canadian Journal of Mathematics.

[6]  Robert L. Bryant,et al.  Conformal and minimal immersions of compact surfaces into the 4-sphere , 1982 .

[7]  Jean-Michel Coron,et al.  Multiple solutions of H‐systems and Rellich's conjecture , 1984 .

[8]  J. Eells,et al.  Selected topics on harmonic maps , 1983 .

[9]  Edmund Taylor Whittaker,et al.  On the General Solution of Laplace's Equation and the Equation of Wave Motions, and on an undulatory explanation of Gravity , 1902 .

[10]  Robert Hardt,et al.  The singular set of an energy minimizing map fromB4 toS2 , 1990 .

[11]  Martin Fuchs,et al.  p-Harmonic obstacle problems , 1990 .

[12]  Jean-Michel Coron,et al.  A Cohomological Criterion for Density of Smooth Maps in Sobolev Spaces Between Two Manifolds , 1991 .

[13]  Michael Struwe,et al.  Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems , 1990 .

[14]  Jean-Michel Coron,et al.  Relaxed Energies for Harmonic Maps , 1990 .

[15]  Shiing-Shen Chern,et al.  Minimal Surfaces in an Euclidean Space of N Dimensions , 1965 .

[16]  J. Coron,et al.  Explosion en temps fini pour le flot des applications harmoniques , 1989 .

[17]  François Murat.  Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant , 1981 .

[18]  V. Benci,et al.  The Dirichlet Problem for Harmonic Maps from the Disk into the Euclidean n-Sphere. , 1985 .

[19]  E. Stein.  On the theory of harmonic functions of several variables , 1961 .

[20]  Toru Ishihara,et al.  A mapping of Riemannian manifolds which preserves harmonic functions , 1979 .

[21]  Jalal Shatah,et al.  Weak solutions and development of singularities of the SU(2) σ‐model , 1988 .

[22]  Jürgen Jost,et al.  Two-dimensional geometric variational problems , 1991 .

[23]  Brian White,et al.  Infima of energy functionals in homotopy classes of mappings , 1986 .

[24]  F. Helein.  Difféomorphismes harmoniques entre un ouvert de R3 et une variété riemannienne , 1989 .

[25]  Hlawka.  The calculus of variations in the large , 1939 .

[26]  Andrea Ratto,et al.  Existence and non-existence results for harmonic maps between spheres , 1990 .

[27]  Jürgen Jost,et al.  The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values , 1984 .

[28]  Simon Salamon,et al.  Twistorial construction of harmonic maps of surfaces into four-manifolds , 1985 .

[29]  O. Perron,et al.  Eine neue Behandlung der ersten Randwertaufgabe fürΔu=0 , 1923 .

[30]  James Eells,et al.  Restrictions on harmonic maps of surfaces , 1976 .

[31]  Frédéric Hélein,et al.  Regularity of weakly harmonic maps from a surface into a manifold with symmetries , 1991 .

[32]  Yunmei Chen,et al.  The weak solutions to the evolution problems of harmonic maps , 1989 .

[33]  Robert Gulliver,et al.  Minimizing p-harmonic maps into spheres. , 1989 .

[34]  Willi Jäger,et al.  Uniqueness and stability of harmonic maps and their Jacobi fields , 1979 .

[35]  Jean-Michel Coron,et al.  Large solutions for harmonic maps in two dimensions , 1983 .

[36]  Henri Poincaré,et al.  Sur les Equations aux Derivees Partielles de la Physique Mathematique , 1890 .

[37]  David Kinderlehrer,et al.  Existence and partial regularity of static liquid crystal configurations , 1986 .

[38]  J. Rawnsley,et al.  Twistor Theory for Riemannian Symmetric Spaces: With Applications to Harmonic Maps of Riemann Surfaces , 1990 .

[39]  Elliott H. Lieb,et al.  Co-Area, Liquid Crystals, and Minimal Surfaces , 1988 .

[40]  Charles B. Morrey,et al.  The Problem of Plateau on a Riemannian Manifold , 1948 .

[41]  Karen Uhlenbeck,et al.  A regularity theory for harmonic maps , 1982 .

[42]  Paul Baird,et al.  A conservation law for harmonic maps , 1981 .

[43]  Ding Weiyue,et al.  Lusternik-Schnirelmann theory for harmonic maps , 1986 .

[44]  J. Eells,et al.  Another Report on Harmonic Maps , 1988 .

[45]  S. S. Cairns,et al.  Differential and combinatorial topology : a symposium in honor of Marston Morse , 1965 .

[46]  Richard S. Hamilton,et al.  Harmonic Maps of Manifolds with Boundary , 1975 .

[47]  Luc Lemaire,et al.  Applications harmoniques de surfaces riemanniennes , 1978 .

[48]  Michael Wiegner.  Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme , 1976 .

[49]  Francis E. Burstall.  Harmonic Maps of Finite Energy From Non-Compact Manifolds , 1984 .

[50]  F. Béthuel,et al.  Density of Smooth Functions between Two Manifolds in Sobolev Spaces , 1988 .

[51]  T. Aubin.  Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire , 1976 .

[52]  Henry C. Wente.  An existence theorem for surfaces of constant mean curvature , 1969 .

[53]  Jean-Michel Coron,et al.  Nonuniqueness for the heat flow of harmonic maps , 1990 .

[54]  Jürgen Jost,et al.  Morse-Conley theory for minimal surfaces of varying topological type , 1990 .

[55]  Nicolaos Kapouleas.  Complete constant mean curvature surfaces in euclidean three-space , 1990 .

[56]  Luc Tartar,et al.  The Compensated Compactness Method Applied to Systems of Conservation Laws , 1983 .

[57]  Jürgen Jost,et al.  Existence proofs for harmonic mappings with the help of a maximum principle , 1983 .

[58]  A. Soyeur,et al.  The Dirichlet problem for harmonic maps from the disc into the 2-sphere , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[59]  Karen Uhlenbeck,et al.  Harmonic maps; a direct method in the calculus of variations , 1970 .

[60]  F. Béthuel,et al.  The approximation problem for Sobolev maps between two manifolds , 1991 .

[61]  F. Lin,et al.  A remark on the map x/[x] , 1987 .

[62]  Richard Schoen,et al.  Analytic Aspects of the Harmonic Map Problem , 1984 .

[63]  Andrea Ratto,et al.  HARMONIC MAPS OF SPHERES AND THE HOPF CONSTRUCTION , 1989 .

[64]  Michael Grüter,et al.  Regularity of weak H-surfaces. , 1981 .

[65]  J. Eells,et al.  A Report on Harmonic Maps , 1978 .

[66]  Henry C. Wente.  Counterexample to a conjecture of H. Hopf , 1986 .

[67]  L. Simon.  Asymptotics for a class of non-linear evolution equations, with applications to geometric problems , 1983 .

[68]  Shing-Tung Yau,et al.  Existence of incompressible minimal surfaces and the topology of three - dimensional manifolds with , 1979 .

[69]  Hermann Karcher,et al.  Non-existence results and growth properties for harmonic maps and forms. , 1984 .

[70]  Haim Brezis,et al.  Regularity of minimizers of relaxed problems for harmonic maps , 1991 .

[71]  J. Sacks,et al.  The existence of minimal immersions of two-spheres , 1977 .

[72]  F. Béthuel,et al.  A characterization of maps in $H^1 (B^3, S^2)$ which can be approximated by smooth maps , 1990 .

[73]  P. Lions.  The concentration-compactness principle in the Calculus of Variations , 1984 .

[74]  Michael Struwe,et al.  On the evolution of harmonic maps in higher dimensions , 1988 .

[75]  J. Sacks,et al.  The Existence of Minimal Immersions of 2-Spheres , 1981 .

[76]  Roger Penrose,et al.  Solutions of the Zero-Rest-Mass Equations , 1969 .

[77]  G D Birkhoff,et al.  Dynamical Systems with Two Degrees of Freedom. , 1917, Proceedings of the National Academy of Sciences of the United States of America.

[78]  Michael Struwe,et al.  Existence and partial regularity results for the heat flow for harmonic maps , 1989 .

[79]  Stephan Luckhaus,et al.  Partial hölder continuity for minima of certain energies among maps into a Riemannian manifold , 1988 .

[80]  E. Stein,et al.  Hp spaces of several variables , 1972 .

[81]  Sagun Chanillo,et al.  Sobolev inequalities involving divergence free maps , 1991 .

[82]  Stefan Hildebrandt,et al.  On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order , 1977 .

[83]  E. Ruh,et al.  The tension field of the Gauss map , 1970 .

[84]  Karen Uhlenbeck,et al.  Regularity of minimizing harmonic maps into the sphere , 1984 .

[85]  L. Schwartz.  Théorie des distributions , 1966 .

[86]  Jean-Michel Coron,et al.  Harmonic maps with defects , 1986 .

[87]  Chang Kungching,et al.  Heat flow and boundary value problem for harmonic maps , 1989 .

[88]  Wei Yei Ding,et al.  Symmetric harmonic maps between spheres , 1988 .

[89]  R. T. Smith,et al.  HARMONIC MAPPINGS OF SPHERES , 1972 .

[90]  Eugenio Calabi,et al.  Minimal immersions of surfaces in Euclidean spheres , 1967 .

[91]  H. Bateman,et al.  The Solution of Partial Differential Equations by Means of Definite Integrals , 1904 .

[92]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[93]  J. Rawnsley,et al.  F-structures, F-twistor spaces and harmonic maps , 1985 .

[94]  Enrico Giusti,et al.  On the regularity of the minima of variational integrals , 1982 .

[95]  Stefan Hildebrandt,et al.  Harmonic mappings into Riemannian manifolds with non-positive sectional curvature. , 1975 .

[96]  Robert Gulliver,et al.  A Theory of Branched Immersions of Surfaces , 1973 .

[97]  Karen Uhlenbeck,et al.  Boundary regularity and the Dirichlet problem for harmonic maps , 1983 .

[98]  Brian White,et al.  Homotopy classes in Sobolev spaces and energy minimizing maps , 1985 .

[99]  Karen K. Uhlenbeck.  Harmonic maps into Lie groups: classical solutions of the chiral model , 1989 .

[100]  N. J. Hitchin,et al.  Harmonic maps from a 2-torus to the 3-sphere , 1990 .

[101]  Mariano Giaquinta,et al.  A priori estimates for harmonic mappings. , 1982 .

[102]  F. Burstall,et al.  Non-linear functional analysis and harmonic maps , 1984 .

[103]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[104]  Willi Jäger,et al.  Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems. , 1983 .

[105]  F. John,et al.  On functions of bounded mean oscillation , 1961 .

[106]  Hermann Weyl,et al.  The method of orthogonal projection in potential theory , 1940 .

[107]  Lawrence C. Evans,et al.  Partial regularity for stationary harmonic maps into spheres , 1991 .

[108]  Jürgen Jost,et al.  A maximum principle for harmonic mappings which solve a Dirichlet problem , 1982 .

[109]  J. Eells,et al.  Harmonic Mappings of Riemannian Manifolds , 1964 .

[110]  Ulrich Pinkall,et al.  On the classification of constant mean curvature tori , 1989 .

[111]  Jürgen Jost,et al.  On the existence of harmonic diffeomorphisms between surfaces , 1982 .

[112]  Richard S. Palais,et al.  The principle of symmetric criticality , 1979 .

[113]  Alexander I. Bobenko,et al.  All constant mean curvature tori inR3,S3,H3 in terms of theta-functions , 1991 .