Bulletin of the American Mathematical Society

Author(s): Robert M. Hardt
Journal: Bull. Amer. Math. Soc. 34 (1997), 15-34.
MSC (1991): Primary 58E20; Secondary 35J45, 35J50
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Abstract: This article surveys research on the existence, structure, behavior, and asymptotics of singularities of harmonic maps.

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31.: Duzaar, F., Grotowski, J.F.: A mixed boundary value problem for energy minimizing harmonic maps, Math. Z. 221 (1996), no. 1, 153-167. MR 1:369 467
32.: Duzaar, F., Steffen, K.: A partial regularity theorem for harmonic maps at a free boundary, Asymptotic Anal. 2 (1989), no. 4, 299-343. MR 90m:58041
33.: Duzaar, F., Steffen, K.: An optimal estimate for the singular set of a harmonic map in the free boundary, J. Reine Angew. Math. 401 (1989), 157-187. MR 90m:58042
34.: Eells, J., Lemaire, L.: A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1-68. MR 82b:58033
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36.: Eells, James, Jr.; Sampson, J. H.: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. MR 29:1603
37.: Evans, L.C.: Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), no.2, 101-113. MR 93m:58026
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42.: Fuchs, M.: -harmonic obstacle problems, II. Extensions of maps and applications, Manuscripta Math. 63 (1989), no. 4, 381-419. MR 91m:49045
43.: Giaquinta, M., Giusti, E.: The singular set of the minima of certain quadratic functionals, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 1, 45-55. MR 86a:49086
44.: Giaquinta, M., Modica, G., Sou\v{c}ek, J.: The Dirichlet energy of mappings with values into the sphere. Manuscripta Math. 65 (1989), no. 4, 489-507. MR 90i:49053
45.: Giaquinta, M.; Modica, G.; Sou\v{c}ek, J.: Cartesian currents and variational problems for mappings into spheres, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), no. 3, 393-485 (1990). MR 91f:49053
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49.: Gulliver, R., Jost, J.: Harmonic maps which solve a free-boundary problem, J. Reine Angew. Math. 381 (1987), 61-89. MR 89k:58067
50.: Gulliver, R., White, B.: The rate of convergence of a harmonic map at a singular point, Math. Ann. 283 (1989), no. 4, 539-549. MR 90j:58034
51.: Hamilton, R.S.: Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics, 471, Springer-Verlag, Berlin-New York, 1975. i+168 pp. MR 58:2872
52.: Hardt, R., Kinderlehrer, D.: Some regularity results in ferromagnetism, preprint.
53.: Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys. 105 (1986), no. 4, 547-570. MR 88a:35207
54.: Hardt, R.; Kinderlehrer, D.; Lin, F.-H.: Stable defects of minimizers of constrained variational principles, Ann. Inst. H. Poincaré Anal. Non Lineaire 5 (1988), no. 4, 297-322. MR 90c:49039
55.: Hardt, R., Kinderlehrer, D., Lin, F.-H.: The variety of configurations of static liquid crystals, Variational methods (Paris, 1988), 115-131, Progr. Nonlinear Differential Equations Appl., 4, Birkhauser Boston, Boston, MA, 1990. MR 94h:58061
56.: Hardt, R., Lin, F.H.: A remark on mappings, Manuscripta Math. 56 (1986), no. 1, 1-10. MR 87k:58068
57.: Hardt, R., Lin, F.H.: Mappings minimizing the norm of the gradient, Comm. Pure Appl. Math. 40 (1987), no. 5, 555-588. MR 88k:58026
58.: Hardt, R., Lin, F.-H.: Stability of singularities of minimizing harmonic maps, J. Differential Geom. 29 (1989), no. 1, 113-123. MR 90a:58037
59.: Hardt, R., Lin, F.H.: Partially constrained boundary conditions with energy minimizing mappings, Comm. Pure Appl. Math. 42 (1989), no. 3, 309-334. MR 90b:49060
60.: Hardt, R., Lin, F.H.: The singular set of an energy minimizing map from $B\sp 4$ to $S\sp 2$ , Manuscripta Math. 69 (1990), no. 3, 275-289. MR 91m:58035
61.: Hardt, R., Lin, F.H.: Harmonic maps into round cones and singularities of nematic liquid crystals, Math. Z. 213 (1993), no. 4, 575-593. MR 94h:58062
62.: Hardt, R., Lin, F.-H.: Singularities for -energy minimizing unit vectorfields on planar domains, Calc. Var. 3 (1995), 311-341.
63.: Hardt, R., Lin, F.-H.: Higher dimensional singularities of -harmonic maps. In preparation.
64.: Hardt, R., Lin, F.H., Poon, C.: Axially symmetric harmonic maps minimizing a relaxed energy, Comm. Pure Appl. Math. 45 (1992), 417-459. MR 93c:58054
65.: Hardt, R., Lin, F.-H., Wang, C.Y.: Singularities for -energy minimizing maps, preprint.
66.: Hardt, R., Lin, F.-H., Wang, C.Y.: The p-energy minimality of , preprint.
67.: Hélein, F.: Regularité des applications faiblement harmoniques entre une surface et une sphere, C. R. Acad. Sci. Paris Ser. I Math. 311 (1990), no. 9, 519-524. MR 92a:58034
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75.: Korevaar, N.J., Schoen, R,M.: Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561-659. MR 95b:58043
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77.: Liao, G.: A regularity theorem for harmonic maps with small energy, J. Diff. Geom. 22 (1985), 233-241. MR 87f:58043
78.: Lin, F.-H.: A remark on the map $x/\vert x\vert$ , C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), no. 12, 529-531. MR 89b:58056
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80.: Lin, F.-H.: Static and moving vortices in Ginzburg-Landau theories, Barrett Lecture Series, Univ. Tennessee, March, 1995.
81.: Lin, F.H.: Gradient estimate and blowup analysis for stationary harmonic maps, In preparation.
82.: Lin, F.-H.: Rectifibility of defect measures and its applications, I. Mapping problems, II. Line vortices of Ginzburg-Landau Equations. In preparation.
83.: Luckhaus, S.: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J. 37 (1988), no. 2, 349-367. MR 89m:58043
84.: Luckhaus, S.: Convergence of minimizers for the -Dirichlet integral, Math. Z. 213 (1993), no. 3, 449-456. MR 94d:58041
85.: Morrey, C.B.: The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807-851. MR 10:259f
86.: Mou, L.H.: Uniqueness of energy minimizing maps for almost all smooth boundary data, Indiana Univ. Math. J. 40 (1991), no. 1, 363-392. MR 92e:58056
87.: Mou, L., Yang, P.: Regularity of -harmonic maps, To appear in J. Geom. Anal.
88.: Musina, R.: Lower bounds for the -energy and a minimization property of the map $x/\vert x\vert$ , Ricerche Mat. 43 (1994), no. 2, 335-346. MR 96a:58071
89.: Poon, C.-C.: Some new harmonic maps from to , J. Differential Geom. 34 (1991), no. 1, 165-168. MR 92e:58057
90.: Preiss, D.: Geometry of measures in $% \textbf { R}^n$ : distribution, rectifiability, and densities, Ann. of Math. (2) 125 (1987), no. 3, 537-643. MR 88d:28008
91.: Reifenberg, E.R.: Solution of the Plateau Problem for -dimensional surfaces of varying topological type. Acta Math. 104 (1960), 1-92. MR 22:4972
92.: Rivière, T.: Harmonic maps from into having a line of singularities, C.R. Acad. Sci. Paris Sér. I, Math. 313 (1991), 583-587. MR 93a:58050
93.: Rivière, T.: Everywhere discontinuous harmonic maps into spheres, Acta Math. 175 (1995), no. 2, 197-226. MR 1:368 247
94.: Rivière, T.: Line Vortices in the Higgs model, preprint.
95.: Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981) 1-24. MR 82f:58035
96.: Schoen, R.M.: Analytic aspects of the harmonic map problem. Seminar on nonlinear partial differential equations, 321-358, MSRI Publ., 2, Springer, New York-Berlin, 1984. MR 86b:58032
97.: Schoen, R.S., Uhlenbeck, K.: A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307-335; 18 (1983), 329. MR 84b:58037a; MR 84b:58037b
98.: Schoen, R.S., Uhlenbeck, K.: Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18 (1983), 253-268. MR 85b:58037
99.: Schoen, R., Uhlenbeck, K.: Regularity of minimizing harmonic maps into the sphere, Invent. Math. 78 (1984), no. 1, 89-100. MR 86a:58024
100.: Serbinowski, T.: Boundary regularity of harmonic maps to nonpositively curved metric spaces. Comm. Anal. Geom. 2 (1994), no. 1, 139-153. MR 95k:58050
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102.: Simon, L.: Isolated singularities of extrema of geometric variational problems. Harmonic mappings and minimal immersions (Montecatini, 1984), 206-277, Lecture Notes in Math., 1161, Springer, Berlin-New York, 1985. MR 87d:58045
103.: Simon, Leon: Proof of the basic regularity theorem for harmonic maps. Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), 225-256, IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, 1996. MR 1:369 590
104.: Simon, L.: Cylindrical tangent cones and the singular set of minimal submanifolds. J. Differential Geom. 38 (1993), no. 3, 585-652. MR 95a:58026
105.: Simon, L.: Uniqueness of some cylindrical tangent cones. Comm. Anal. Geom. 2 (1994), no. 1, 1-33. MR 95m:49026
106.: Simon, L.: Rectifiability of the singular set of energy minimizing maps, Calc. of Var. P.D.E. 3 (1995), 1-65.
107.: Simon, L.: Theorems on the regularity and singularity of minimal surfaces and harmonic maps. Geometry and global analysis, Tohoku Univ., Sendai, 1993, 111-145. MR 1:361 175
108.: Simon, L.: Theorems on regularity and singularity of energy minimizing maps, ETH Lectures in Mathematics Series, Birkhauser, 1996.
109.: Simon, L.: On the singularities of harmonic maps, In preparation.
110.: Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), no. 4, 558-581. MR 87e:58056
111.: Struwe, M.: On the asymptotic behavior of minimizers of the Ginzburg-Landau model in dimensions, Differential Integral Equations 7 (1994), no. 5-6, 1613-1624. MR 95g:35057a
112.: Tolksdorf, P.: Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl. (4) 134 (1983), 241-266. MR 85h:35104
113.: Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3-4, 219-240. MR 57:14031
114.: Wang, C.Y.: Energy minimizing maps to polyhedra, preprint.
115.: Wang, C.Y.: Minimality, stability, and perturbation for some p-harmonic maps, preprint.
116.: Wang, S.S.: -energy minimizing sections of a fiber bundle. To appear in Mich. Math. J.
117.: Wei, S.W.: -harmonic maps and their applications to geometry, topology, and analysis, preprint.
118.: Wei, S.W., Yau, C.-M.: Regularity of -energy minimizing maps and -superstrongly unstable indices, J. Geom. Anal. 4 (1994), no. 2, 247-272. MR 95f:58028
119.: White, B.: Nonunique tangent maps at isolated singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 1, 125-129. MR 92e:58058
120.: Wolf, M.: Harmonic maps from surfaces to $\mathbf R$ -trees, Math. Z. 218 (1995), no. 4, 577-593. MR 1:326 987
121.: Zhang, D.: The existence of nonminimal regular harmonic maps from to , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), no. 3, 355-365 (1990). MR 91c:58036

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