The Evolution of Harmonic Maps: Existence, Partial Regularity, and Singularities

We survey recent results on global existence and partial regularity for the heat flow for harmonic maps. A new proof of short-time blow-up in dimensions m ≥3 is given.

[1]  J. Moser.  A Harnack inequality for parabolic di2erential equations , 1964 .

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

[3]  M. Struwe.  Heat-flow methods for harmonic maps of surfaces and applications to free boundary problems , 1988 .

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

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

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

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

[8]  J. Keller,et al.  Reaction-diffusion processes and evolution to harmonic maps , 1989 .

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

[10]  Michael Struwe,et al.  On the evolution of harmonic mappings of Riemannian surfaces , 1985 .

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

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

[13]  Jürgen Jost,et al.  Ein Existenzbeweis für harmonische Abbildungen, die ein Dirichletproblem lösen, mittels der Methode des Wärmeflusses , 1981 .

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

[15]  Wei Ding,et al.  Blow-up and global existence for heat flows of harmonic maps , 1990 .

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

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

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

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

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