The analysis of harmonic maps and their heat flows

The Analysis of Harmonic Maps and their Heat Flows

Volumn , Issue , 2008, Pages 1-267

  Lin, Fanghua ^a   Wang, Changyou ^b  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITY OF KENTUCKY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

HEAT TRANSFER; STUDENTS;

EXISTENCE THEOREM; GEOMETRIC ANALYSIS; GEOMETRIC PARTIAL DIFFERENTIAL EQUATION; GLOBAL SMOOTH SOLUTIONS; GRADUATE STUDENTS; HIGHER DIMENSIONS; REGULAR SOLUTION; RIEMANNIAN MANIFOLD;

HARMONIC ANALYSIS;

EID: 84967355712     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/6679     Document Type: Book

References (215)

1. W. Allard, An Integrality Theorem and A Regularity Theorem for surfaces whose First Vaiation with respect to a Parametric Elliptic Integrals is Controlled, Proceeding of Symp. In Pure Math. Vol. 44 (1986), AMS. 1-2
2. W. Allard, On the first variation of a varifold. Ann. of Math. (2) 95 (1972), 417-491.
3. W. Allard, F. Almgren, On the radial behavior of minimal surfaces and the uniqueness of their tangent cones. Ann. of Math. (2) 113 (1981), no. 2, 215-265.
4. F. Almgren, Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension 2. Preprint, Princeton University (1984).
5. F. Almgren, The theory of varifolds. Mimeographed Notes. Princeton. 1965.
6. F. Almgren, E. Lieb, Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples, and bounds. Ann. of Math. (2) 128 (1988), no. 3, 483-530.
7. L. Ambrosio, H. M. Soner, A measure -theoretic approach to higher codimension mean curvature ows. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 27-49 (1998).
8. L. Ambrosio, H. M. Soner, Level set approach to mean curvature ow in arbitrary codimension. J. Difi. Geom. 43 (4) (1996), 693-737.
9. G. Dell Antonio, D. Zwanziger, Every gauge orbit passes inside the Gribov horizon. Comm. Math. Phys. 138 (1991), no. 2, 291-299.
10. F. Bethuel, The approximation problem for the Sobolev maps between manifolds. Acta Math. 167 (1991), 167-201.
11. F. Bethuel, On the singular set of stationary harmonic maps. Manu. Math. 78 (1993), 417-443.
12. F. Bethuel, J. Coron, J. Ghidaglia, A. Soyeur, Heat ows and relaxed energies for harmonic maps. Nonlinear difiusion equations and their equilibrium states, 3 (Gregynog, 1989), 99-109, Progr. Nonlinear Differential Equations Appl., 7, Birkhäuser Boston, Boston, MA, 1992
13. M. Bertsch, R. Dal Passo, R. Van der Hout, Nonuniqueness for the heat ow of harmonic maps on the disk. Arch. Ration. Mech. Anal. 161 (2002), no. 2, 93-112.
14. F. Bethuel, H. Brezis, J. Coron, Relaxed energies for harmonic maps. Variational methods (Paris, 1988), 37-52, Progr. Nonlinear Differential Equations Appl., 4, Birkhuser Boston, Boston, MA, 1990.
15. K. Brakke, The motion of a surface by its mean curvature. Mathematical Notes, 20. Princeton University Press, Princeton, N.J., 1978.
16. k-valued maps with singularities. Topics in calculus of variations (Montecatini Terme, 1987), 1-30, Lecture Notes in Math., 1365, Springer, Berlin, 1989.
17. H. Brezis, The interplay between analysis and topology in some nonlinear PDE problems. Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 2, 179-201.
18. H. Brezis and J. Coron, Convergence of solutions of H-systems or how to blow bubbles. Arch. Rational Mech. Anal. 89 (1985), no. 1, 21-56.
19. H. Brezis, J. Coron, E. Lieb, Harmonic maps with defects. Comm. Math. Phys. 107 (1986), 649-705.
20. L. Cafiarelli, Regularity theorems for weak solutions of some nonlinear systems. Comm. Pure Appl. Math. 35 (1982), no. 6, 833-838.
21. L. Cafiarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771-831.
22. K. C. Chang, W. Y. Ding, A result on the global existence for heat ows of harmonic maps from D2 into S2. Nematics (Orsay, 1990), 37-47, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 332, Kluwer Acad. Publ., Dordrecht, 1991.
23. K. C. Chang, W. Y. Ding, R. Ye, Finite-time blow-up of the heat ow of harmonic maps from surfaces. J. Differential Geom. 36 (1992), no. 2, 507-515.
24. S. A. Chang, L. Wang, P. Yang, Regularity of harmonic maps. Comm. Pure Appl. Math. 52 (1999), no. 9, 1099-1111.
25. 2,λ). Ann. Mat. Pura Appl. (4) 69 (1965), 321-381.
26. X. Cheng, Estimate of the singular set of the evolution problem for harmonic maps, J. Differential Geom. 34 (1991), no. 1, 169-174.
27. Y. Chen, The weak solutions to the evolution problems of harmonic maps, Math. Z., 201 (1989), 69-74.
28. Y. M. Chen, W. Y. Ding, Blow-up and global existence for heat ows of harmonic maps. Invent. Math. 99 (1990), no. 3, 567-578.
29. Y. Chen, J. Li, F. H. Lin, Partial regularity for Weak heat ows into Sphere. Comm. Pure Applied Math. 48 (1995), no. 4, 429-448.
30. Y. M. Chen, F. H. Lin, Remarks on approximate harmonic maps. Comm. Math. Helv. 70 (1995), no.1, 161-169.
31. Y. M. Chen, F. H. Lin, Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1 (1993), no. 3-4, 327-346.
32. Y. M. Chen, F. H. Lin, Evolution equations with a free boundary condition. J. Geom. Anal. 8 (1998), no. 2, 179-197.
33. Y. M. Chen, M. Struwe, Existence and partial regularity for heat ow for harmonic maps. Math.Z. 201 (1989) 83-103.
34. Y. M. Chen, C. Y. Wang, Partial regularity for weak heat ows into Riemannian homogeneous spaces. Comm. Partial Difi. Equations, 21 (1996), 735-761.
35. R. Coifman, P. L. Lions, Y. Meyers, S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72 (1993), no. 3, 247-286.
36. K. C. Chang, Heat ow and boundary value problem for harmonic maps. Ann. Inst. H. Poincar Anal. Non Linéaire 6 (1989), no. 5, 363-395.
37. K. Corlette, Harmonic maps, rigidity, and Hodge theory. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 465-471, Birkhäuser, Basel, 1995.
38. J. Coron, Nonuniqueness for the heat ow of harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linfiaire 7 (1990), no. 4, 335-344.
39. J. Coron, J. Ghidaglia, Explosion en temps fini pour le ot des applications harmoniques (French) [Finite-time blow-up for the heat ow of harmonic mappings]. C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 12, 339-344.
40. J. Coron, R. Gulliver, Minimizing p-harmonic maps into spheres. J. Reine Angew. Math. 401 (1989), 82-100.
41. W. Y. Ding, Lectures on the Heat Flow of Harmonic Maps. Lecture Notes at CTS, NTHU, Taiwan (1998).
42. W. Y. Ding, F. H. Lin, A generalization of Eells-Sampson’s theorem. J. Partial Differential Equations 5 (1992), no. 4, 13-22.
43. W. Y. Ding, G. Tian, Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3 (1995), no. 3-4, 543-554.
44. F. Duzaar, E. Kuwert, Minimization of conformally invariant energies in homotopy classes. Calc. Var. Partial Differential Equations 6 (1998), no. 4, 285-313.
45. L. C. Evans, Partial regularity for stationary harmonic maps into spheres. Arch. Rat. Mech. Anal. 116 (1990) 203-218.
46. L. C. Evans, Weak convergence methods for nonlinear partial differential equations. CBMS Regional Conference Series in Mathematics, 74. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990.
47. L. C. Evans, Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
48. L. C. Evans, R. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.
49. J. Eells, J. Sampson, Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 1964 109-160.
50. J. Eells, L. Lemaire, A report on harmonic maps. Bull. London Math. Soc. 10 (1978), 1-68.
51. J. Eells, L. Lemaire, Another report on harmonic maps. Bull. London Math. Soc. 20 (1988), 385-524.
52. J. Eells, J. C. Wood, Restrictions on harmonic maps of surfaces. Topology 15 (1976), no. 3, 263-266
53. L. Escauriaza, G. Seregin, V. Sverfiak, Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169 (2003), no. 2, 147-157.
54. H. Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing at chains modulo two with arbitrary codimension. Bull. Amer. Math. Soc. 76 (1970), 767-771.
55. H. Federer, Geometric Measure Theory. Springer-Verlag, New York Inc. 1969.
56. H. Federer, W. P. Ziemer, The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana Univ. Math. J. 22 (1972/73), 139-158.
57. C. Feffrman, E. Stein, Hp spaces of several variables. Acta Math. 129 (1972), 137-193.
58. M. Feldman, Partial regularity for harmonic maps of evolution into spheres. Comm. Partial Differential Equations 19 (1994), no. 5-6, 761-790.
59. A. Freire, Uniqueness for the harmonic map ow in two dimensions. Calc. Var. Partial Differential Equations 3 (1995), no. 1, 95-105.
60. A. Freire, Uniqueness for the harmonic map ow from surfaces to general targets. Comment. Math. Helv. 70 (1995), no. 2, 310-338.
61. M. Fuchs, p-harmonic obstacle problems. I. Partial regularity theory. Ann. Mat. Pura Appl. (4) 156 (1990), 127-158.
62. M. Grayson, R. Hamilton, The formation of singularities in the harmonic map heat ow. Comm. Anal. Geom. 4 (1996), no. 4, 525-546.
63. S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry. Second edition. Universitext. Springer-Verlag, Berlin, 1990.
64. R. Gulliver, B. White, The rate of convergence of a harmonic map at a singular point. Math. Ann. 283 (1989), no. 4, 539-549.
65. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ, 1983.
66. M. Giaquinta, E. Giusti, The singular set of the minima of certain quadratic functionals. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 1, 45-55.
67. M. Giaquinta, E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148 (1982), 31-46.
68. M. Giaquinta, S. Hildebrandt, A priori estimates for harmonic mappings. J. Reine Angew. Math. 336 (1982), 124-164.
69. M. Giaquinta, G. Modica, J. Souček, The Dirichlet energy of mappings with values into the sphere. Manuscripta Math. 65 (1989), no. 4, 489-507.
70. M. Giaquinta, G. Modica, J. Souček, Cartesian currents in the calculus of variations. I. Cartesian currents. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 37. Springer-Verlag, Berlin, 1998.
71. M. Giaquinta, G. Modica, J. Souficek, Cartesian currents in the calculus of variations. II. Variational integrals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 38. Springer-Verlag, Berlin, 1998.
72. D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, 1997.
73. M. Gromov, Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307-347.
74. M. Gromov, R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. № 76 (1992), 165-246.
75. M. Grüter, Conformally invariant variational integrals and the removability of isolated singularities. Manuscripta Math. 47 (1984), no. 1-3, 85-104.
76. G. Huisken, Asymptotic behavior for singularities of the mean curvature ow. J. Differential Geom. 31 (1990), no. 1, 285-299.
77. R. Hamilton, Harmonic maps of manifolds with boundary. Lecture Notes in Mathematics, Vol. 471. Springer-Verlag, Berlin-New York, 1975.
78. Z. C. Han, L. F. Tam, A. Treibergs, T. Wan, Harmonic maps from the complex plane into surfaces with nonpositive curvature. Comm. Anal. Geom. 3 (1995), no. 1-2, 85-114.
79. F. Hang, F. H. Lin, Topology of Sobolev mappings. II. Acta Math. 191 (2003), no. 1, 55-107.
80. R. Hardt, Singularities of harmonic maps. Bull. Amer. Math. Soc. (N.S.) 34(1997), no. 1, 15-34.
81. R. Hardt, Singularities of harmonic maps. Nonlinear partial differential equations, Proceedings of International Conference, Zhejiang University, June 1992, 74-80. International Academic, Beijing, 1993.
82. R. Hardt, F. H. Lin, C. Y. Wang, Singularities of p-energy minimizing maps. Comm. Pure Appl. Math. 50 (1997), no. 5, 399-447.
83. p norm of the gradient. Comm. Pure Appl. Math. 40 (1987), no. 5, 555-588.
84. R. Hardt, F. H. Lin, Stability of singularities of minimizing harmonic maps. J. Differential Geom. 29 (1989), no. 1, 113-123.
85. R. Hardt, F. H. Lin, The singular set of an energy minimizing map from B4 to S2. Manuscripta Math. 69 (1990), no. 3, 275-289.
86. P. Hartman, On homotopic harmonic maps. Can. J. Math. 19 (1967), 637-687.
87. E. Heinz, On Wyel ‘s embedding problem. J.Math.Mech.11(1962), 421-454.
88. E. Heinz, Existence theorems for one-to-one mappings associated with elliptic systems of second order, II. Jour.d’Analyse Mathematique 17(1966),145-184.
89. F. Hélein, Harmonic diffeomorphisms between Riemannian manifolds. Variational methods (Paris, 1988), 309-318, Progr. Nonlinear Differential Equations Appl., 4, Birkhuser Boston, Boston, MA, 1990.
90. F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une sphére. (French) [Regularity of weakly harmonic maps between a surface and an n-sphere] C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 9, 519-524.
91. F. Hélein, Regularity of weakly harmonic maps from a surface into a manifold with symmetries. Manuscripta Math. 70 (1991), no. 2, 203-218.
92. F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. (French) [Regularity of weakly harmonic maps between a surface and a Riemannian manifold] C. R. Acad. Sci. Paris Sér. I Math. 312 (1991)
93. F. Hélein, Harmonic maps, conservation laws and moving frames. Translated from the 1996 French original. With a foreword by James Eells. Second edition. Cambridge Tracts in Mathematics, 150. Cambridge University Press, Cambridge, 2002.
94. F. Hélein, J. C. Wood, Harmonic maps, in Handbook of Global Analysis, D. Krupka and D. Saunders, editors, Elsevier, 2008, p. 417-491.
95. S. Hildebrandt, H. Kaul, K. Widman, An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138 (1977), no. 1-2, 1-16.
96. M. C. Hong, Some new examples for nonuniqueness of the evolution problem of harmonic maps. Comm. Anal. Geom. 6 (1998), no. 4, 809-818.
97. M. C. Hong, On the Hausdorfi dimension of the singular set of stablestationary harmonic maps. Comm. Partial Differential Equations 24 (1999), no. 11-12, 1967-1985.
98. M. C. Hong, C. Y. Wang, On the singular set of stable-stationary harmonic maps. Calc. Var. Partial Differential Equations 9 (1999), no. 2, 141-156.
99. T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. 108 (1994), no. 520.
100. T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differential Geom. 38 (1993), no. 2, 417-461.
101. T. Iwaniec, G. Martin, Quasiregular mappings in even dimensions. Acta Math. 170 (1993), no. 1, 29-81.
102. J. Jost, Harmonic mappings between Riemannian manifolds. Proceedings of the Center for Mathematical Analysis, Australian National University, Vol. 4, 1983.
103. J. Jost, Two dimensional geometric variational problems. New York, Wiley, 1991.
104. J. Jost, Equilibrium maps between metric spaces. Calc. Var. Partial Differential Equations 2 (1994), no. 2, 173-204.
105. J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70 (1995), no. 4, 659-673.
106. J. Jost, S. T. Yau, A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry. Acta Math. 170 (1993), no. 2, 221-254.
107. J. Jost, S. T. Yau, Harmonic maps and superrigidity. Tsing Hua lectures on geometry & analysis (Hsinchu, 1990-1991), 213-246, Int. Press, Cambridge, MA, 1997.
108. W. Jäger, H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems. J. Reine Angew. Math. 343 (1983), 146-161.
109. F. John, L. Nirenberg, On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 1961 415-426.
110. J. Keller, J. Rubinstein, P. Sternberg, Reaction-difiusion processes and evolution to harmonic maps. SIAM J. Appl. Math. 49 (1989), no. 6, 1722-1733.
111. N. Korevaar, R. Schoen, Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1 (1993), no. 3-4, 561-659.
112. N. Korevaar, R. Schoen, Global existence theorems for harmonic maps to nonlocally compact spaces. Comm. Anal. Geom. 5 (1997), no. 2, 333-387.
113. F. Labourie, Elliptic isometric immersions and pseudoholomorphic curves. J. Differential Geom. 30 (1989), no. 2, 395-424.
114. L. Lemaire, Applications harmoniques de surfaces riemanniennes. (French) J. Differential Geom. 13 (1978), no. 1, 51-78.
115. 2 are integrable. J. London Math. Soc. (2) 66 (2002), no. 2, 468-486.
116. P. Li, L. F. Tam, The heat equation and harmonic maps of complete manifolds. Invent. Math. 105 (1991), no. 1, 1-46.
117. P. Li, L. F. Tam, Uniqueness and regularity of proper harmonic maps. Ann. of Math. (2) 137 (1993), no. 1, 167-201.
118. P. Li, L. F. Tam, Uniqueness and regularity of proper harmonic maps. II. Indiana Univ. Math. J. 42 (1993), no. 2, 591-635.
119. P. Li, L. F. Tam, J. P. Wang, Harmonic diffomorphisms between Hadamard manifolds. Trans. Amer. Math. Soc. 347 (1995), no. 9, 3645-3658.
120. G. M. Lieberman, Second order parabolic differential equations. World Scienti fic Publishing Co., Inc., River Edge, NJ, 1996.
121. F. H. Lin, On Darboux’s System, Nonlinear Analysis and Microlocal Analysis. Nankai Series in Pure and Appl.Math. and Theoretical Phys., vol 2(1992), World Scientific, edited by K.C.Chang & T.T.Li.
122. F. H. Lin, Une remarque sure l’s applications x/|x|. C. R. Acad. Sci. Pairs 305 (1987), 529-531.
123. F. H. Lin, Gradient Estimates and Blow-Up Analysis for Stationary Harmonic Maps. Annal Math. 149 (1999), 785-829.
124. F. H. Lin, Mapping problems, fundamental groups and defect measures. Acta Math. Sin. (Engl. Ser.) 15 (1999), no. 1, 25-52.
125. F. H. Lin, Varifold type theory for Sobolev mappings. AMS & IP Stud. Adv. Math., 20, 423-430, Amer. Math. Soc., Providence, RI, 2001.
126. F. H. Lin, A new proof of the Cafiarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51 (1998), no. 3, 241-257.
127. O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor:Leon Ehrenpreis Academic Press, New York-London 1968
128. O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural’ceva, Linear and quasi-linear equations of parabolic type, Moscow 1968.
129. H. Lewy, On the existenc of a closed surface realizing a given Riemannian metric. Proc. Nat. Acad. Sci. USA 24(1938), 104-106.
130. L. Ljusternik, L. Schnirelman, Methodes topologiques dans les problemes variationelles, Actualities Sci. Industr. 188, Paris, 1934.
131. F. H. Lin, T. Riviere, Energy Quantization for Harmonic Maps. Duke Math. Journal 111 (2002), no. 1, 177-193.
132. F. H. Lin, C. Y. Wang, Stable stationary harmonic maps to spheres. Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 319-330.
133. F. H. Lin, C. Y. Wang, Energy identity of harmonic map ows from surfaces at finite singular time. Calc. Var. Partial Differential Equations 6 (1998), no. 4, 369-380.
134. F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres. Comm. Anal. Geom. 7 (1999), no. 2, 397-429.
135. F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres. II. Comm. Anal. Geom. 10 (2002), no. 2, 341-375.
136. F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure and generalized varifold ows. Ann. Inst. H. Poincar Anal. Non Linaire 19 (2002), no. 2, 209-259.
137. J. Y. Li, G. Tian, Blow-up Locus for heat ows of harmonic maps. Acta Math. Sin. (Engl. Ser.) 16 (2000), no.1, 29-62.
138. J. Y. Li, G. Tian, A blow-up formula for stationary harmonic maps. Internat. Math. Res. Notices 1998, no. 14, 735-755.
139. S. Luckhaus, Partial Hlder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J. 37 (1988), no. 2, 349-367
140. S. Luckhaus, Convergence of minimizers for the p-Dirichlet integral. Math. Z. 213 (1993), no. 3, 449-456.
141. L. Ma, Harmonic map heat ow with free boundary. Comment. Math. Helv. 66 (1991), no. 2, 279-301.
142. U. Mayer, Gradient ows on nonpositively curved metric spaces and harmonic maps. Comm. Anal. Geom. 6 (1998), no. 2, 199-253.
143. J. Milnor, Morse Theory, Ann. of Math. Studies 51, Princeton Univ. Press, Princeton 1963.
144. D. McDufi, D. Salamon, J-holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2004. xii+669 pp.
145. C. B. Morrey, Multiple integrals in the calculus of variations. Springer-Verlag, Berlin, 1986.
146. J. Moser, A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964) 101-134.
147. R. Moser, Stationary measures and rectifiability. Calc. Var. Partial Differential Equations 17 (2003), no. 4, 357-368.
148. R. Moser, Regularity for the approximated harmonic map equation and application to the heat ow for harmonic maps. Math. Z. 243 (2003), no. 2, 263-289.
149. T. Nakajima, Singular points of harmonic maps from 4-dimensional domains into 3-spheres. Duke Math. J. 132 (2006), no. 3, 531-543.
150. J. Nash, The imbedding problem for Riemannian manifolds. Annal Math. 63 (1956), 20-63.
151. R. Palais, Foundations of global non-linear analysis. W. A. Benjamin, Inc., New York-Amsterdam 1968.
152. T. Parker, Bubble tree convergences for harmonic maps. J.Difi.Geom. 44(1996) no. 3, 595-633.
153. T. Parker, J. Wolfson, Pseudo-holomorphic maps and bubble trees. J. Geom. Anal. 3 (1993), no. 1, 63-98.
154. D. Preiss, Geometry of measures in Rn: distribution, rectifiability, and densities. Ann. of Math. (2) 125 (1987), no. 3, 537-643.
155. P. Price, A monotonicity formula for Yang-Mills élds. Manuscripta Math. 43 (1983), no. 2-3, 131-166.
156. J. Qing, Boundary regularity of weakly harmonic maps from surfaces. J. Funct. Anal. 114 (1993), no. 2, 458-466.
157. J. Qing, On singularities of the heat ow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3 (1995), no. 1-2, 297-315
158. J. Qing, G. Tian, Bubbling of the heat ows for harmonic maps from surfaces.Comm. Pure Appl. Math. 50 (1997), no. 4, 295-310.
159. E. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type. Acta Math. 104 1960 1-92.
160. T. Riviére, Everywhere discontinuous harmonic maps into spheres. Acta Math. 175 (1995), no. 2, 197-226.
161. T. Riviére, Conservation laws for conformally invariant variational problems. Invent. Math. 168 (2007), no. 1, 1-22.
162. 2. Comm. Anal. Geom. 6 (1998), no. 3, 427-483.
163. T. Riviére, M. Struwe, Partial regularity of stationary harmonic maps, revisited. To appear, Comm. Pure Applied Math.
164. J. Sacks, K. Uhlenbeck, The existence of minimal immersions of 2-spheres. Ann. of Math. (2) 113 (1981), no. 1, 1-24.
165. S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller. Comm. Partial Differential Equations 19 (1994), no. 1-2, 277-319.
166. R. Schoen, Analytic aspects of the harmonic map problem. Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), 321-358, Math. Sci. Res. Inst. Publ., 2, Springer, New York, 1984.
167. R. Schoen, The role of harmonic mappings in rigidity and deformation problems. Complex geometry (Osaka, 1990), 179-200, Lecture Notes in Pure and Appl. Math., 143, Dekker, New York, 1993.
168. R. Schoen, The theory and applications of harmonic mappings between Riemannian manifolds. A lecture presented in Orono, ME, August 1991. AMS Progress in Mathematics Lecture Series. American Mathematical Society, Providence, RI, 1992.
169. R. Schoen, The eéct of curvature on the behavior of harmonic functions and mappings. Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), 127-184, IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, 1996.
170. R. Schoen, L. Simon, Regularity of stable minimal hypersurfaces. Comm. Pure Appl. Math. 34 (1981), no. 6, 741-797.
171. R. Schoen, K. Uhlenbeck, A regularity theory for harmonic mappings. J. Difi. Geom 17 (1982), 307-335.
172. R. Schoen, K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geom. 18 (1983), no. 2, 253-268.
173. R. Schoen, K. Uhlenbeck, Approximation of Sobolev maps between Riemannian manifolds. Preprint (1984).
174. R. Schoen, K. Uhlenbeck, Regularity of minimizing harmonic maps into the sphere. Invent. Math. 78 (1984), no. 1, 89-100.
175. R. Schoen, S. T. Yau, Lectures on differential geometry. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang, Jia Qing Zhong and Yi Chao Xu. Translated from the Chinese by Ding and S. Y. Cheng. Preface translated from the Chinese by Kaising Tso. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994.
176. R. Schoen, S. T. Yau, Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry and Topology, II. International Press, Cambridge, MA, 1997.
177. R. Schoen, S. T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann. of Math. (2) 110 (1979), no. 1, 127-142.
178. R. Schoen, S. T. Yau, Lectures on harmonic maps. Proceedings and Lecture Notes in Geometry and Topology, II. International Press, Cambridge, MA, 1997. vi+394 pp.
179. R. Schoen, L. Simon, S. T. Yau, Curvature estimates for minimal hypersurfaces. Acta Math. 134 (1975), no. 3-4, 275-288
180. J. Shatah, Weak solutions and development of singularities of the SU(2) σ- model. Comm. Pure Appl. Math. 41 (1988), no. 4, 459-469.
181. L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. of Math. (2) 118 (1983), no. 3, 525-571
182. L. Simon, Theorems on the regularity and singularity of minimal surfaces and harmonic maps. Lectures on geometric variational problems (Sendai, 1993), 115-150, Springer, Tokyo, 1996.
183. L. Simon, Isolated singularities of extrema of geometric variational problems. Harmonic mappings and minimal immersions (Montecatini, 1984), 206-277, Lecture Notes in Math., 1161, Springer, Berlin, 1985.
184. L. Simon, Rectifiability of the singular set of energy minimizing maps. Calc. Var. Partial Differential Equations 3 (1995), no. 1, 1-65.
185. L. Simon, Uniqueness of some cylindrical tangent cones. Comm. Anal. Geom. 2 (1994), no. 1, 1-33.
186. L. Simon, Cylindrical tangent cones and the singular set of minimal submanifolds. J. Differential Geom. 38 (1993), no. 3, 585-652.
187. L. Simon, 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
188. L. Simon, Singularities of geometric variational problems. Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), 185-223, IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, 1996.
189. L. Simon, Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983.
190. Y. T. Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems. J. Differential Geom. 17 (1982), no. 1, 55-138.
191. E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ, 1970.
192. E. Stein, Harmonic analysis, Princeton University Press, Princeton 1993.
193. M. Struwe, On the evolution of harmonic mapping of Riemannian surfaces. Comm. Math. Helv. 60 (1985) 558-581
194. M. Struwe, Heat-ow methods for harmonic maps of surfaces and applications to free boundary problems. Partial differential equations (Rio de Janeiro, 1986), 293-319, Lecture Notes in Math., 1324, Springer, Berlin, 1988
195. M. Struwe, On the evolution of harmonic maps in higher dimensions. J. Differential Geom. 28 (1988), no. 3, 485-502.
196. M. Struwe, Geometric evolution problems. Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), 257-339, IAS/Park City Math. Ser., 2, Amer. Math. Soc., Providence, RI, 1996.
197. M. Struwe, The evolution of harmonic mappings with free boundaries. Manu. Math. 70 (1991), no. 4, 373-384.
198. L. Tartar, Imbedding theorems in Lorentz spaces and applications. Preprint (1995).
199. L. F. Tam, T. Y. Wan, Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space. J. Differential Geom. 42 (1995), no. 2, 368-410.
200. G. Tian, Gauge theory and calibrated geometry, I. Ann. Math. 151 (200), 193-268.
201. P. Topping, Reverse bubbling and nonuniqueness in the harmonic map ow. Int. Math. Res. Not. 2002, no. 10, 505-520.
202. P. Topping, Rigidity in the harmonic map heat ow. J. Differential Geom. 45(1997), no. 3, 593-610.
203. P. Topping, Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map ow. Ann. of Math. (2) 159 (2004), no. 2, 465-534.
204. T. Y. Wan, Constant mean curvature surface, harmonic maps, and universal Teichmller space. J. Differential Geom. 35 (1992), no. 3, 643-657.
205. C. Y. Wang, Boundary partial regularity for a class of harmonic maps. Comm. Partial Differential Equations 24 (1999), no. 1-2, 355-368.
206. C. Y. Wang, Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets. Houston J. Math. 22 (1996), no. 3, 559-590.
207. x. To appear, Arch. Ration. Mech. Anal.
208. C. Y. Wang, A remark on harmonic map ows from surfaces. Differential Integral Equations 12 (1999), no. 2, 161-166.
209. B. White, Nonunique tangent maps at isolated singularities of harmonic maps. Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 1, 125-129.
210. B. White, Infima of energy functionals in homotopy classes of mappings. J. Differential Geom. 23 (1986), no. 2, 127-142.
211. B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps. Acta Math. 160 (1988), no. 1-2, 1-17.
212. B. White, Stratification of minimal surfaces, mean curvature ows, and harmonic maps. J. Reine Angew. Math. 488 (1997), 1-35.
213. M. Wolf, The Teichmüller theory of harmonic maps. J. Differential Geom. 29(1989), no. 2, 449-479.
214. S. T. Yau, Some function theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25 (1976), 659-670.
215. W. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989