Subelliptic harmonic maps from Carnot groups

  • Changyou Wang
Original Paper


For subelliptic harmonic maps from a Carnot group into a Riemannian manifold without boundary, we prove that they are smooth near any \(\epsilon\)-regular point (see Definition 1.3) for sufficiently small \(\epsilon > 0\). As a consequence, any stationary subelliptic harmonic map is smooth away from a closed set with zero HQ-2 measure. This extends the regularity theory for harmonic maps (cf. [SU], [Hf], [El], [Bf]) to this subelliptic setting.


Riemannian Manifold Regular Point Regularity Theory Carnot Group 
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  1. 1.
    Barletta, E., Dragomir, S., Urakawa, H.: Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds. Indiana Univ. Math. J. 50, 719--746 (2001)zbMATHGoogle Scholar
  2. 2.
    Bethuel, F.: On the singular set of stationary harmonic maps. Manu. Math. 78, 417--443 (1993)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bony, J.: Principe du maximum inégalité de Harnack et unicité du probléme de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier 19, 227--304 (1969)Google Scholar
  4. 4.
    Coifman, R., Lions, P., Meyers, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. math. Pures. Appl. 72, 699--734 (1993)Google Scholar
  5. 5.
    Capogna, L., Danielli, D., Garofalo, N.: Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations. Amer. J. Math. 118, 1153--1196 (1997)zbMATHGoogle Scholar
  6. 6.
    Capogna, L., Danielli, D., Garofalo, N., The geometric Sobolev embedding for vector fields and the isoperimetric inequality. Comm. Anal. Geom. 2, 203--215 (1994)Google Scholar
  7. 7.
    Citti, G., Garofalo, N., Lanconelli, E.: Harnack's inequality for sum of squares plus potential. Amer. J. Math. 115, 699--734 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Capogna, L., Garofalo, N.: Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type. To appear in J. Eur. Math. Soc.Google Scholar
  9. 9.
    Capogna, L.: Regularity for quasilinear equations and \(1\)-quasiconformal maps in Carnot groups. Math. Ann. 313, 263--295 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Chanillo, S.: Sobolev inequalities involving divergence free maps. Comm. PDE 16, 1969--1994 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. AMS, 83(4), 569--645 (1977)Google Scholar
  12. 12.
    Chang, A., Wang, L., Yang, P.: Regularity of harmonic maps. CPAM 52(9), 1099--1111 (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Evans, L., Gariepy, R.: Measure Theory and Fine properties of Functions. CRC press 1992Google Scholar
  14. 14.
    Evans, L.: Partial regularity for stationary harmonic maps into spheres. Arch. Rat. Mech. Anal. 116, 101--113 (1991)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Eells, J., Sampson, J.: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 85, 109--160 (1964)Google Scholar
  16. 16.
    Folland, G.: A fundamental solution for a subelliptic operator. Bull. AMS 79, 373--376 (1973)zbMATHGoogle Scholar
  17. 17.
    Folland, G.: Subelliptic estimates and function spaces on nilpotent Lie groups. Arkiv. Mat. 13, 161--207 (1975)zbMATHGoogle Scholar
  18. 18.
    Fefferman, C., Phong, D.: Subelliptic eigenvalue problems. Proceedings of Conference on Harmonic Analysis, in honor of A. Zygmund. Wadsworth Math. Series, 590--606 (1981)Google Scholar
  19. 19.
    Fefferman, C., Stein, E.: Hardy spaces of several variables. Acta Math. 129, 137--193 (1972)zbMATHGoogle Scholar
  20. 20.
    Folland, G., Stein, E.: Hardy spaces on homogeneous groups. Math. Notes, vol. 28, Princeton Univ. Press, Princeton, NJ 1982Google Scholar
  21. 21.
    Giaquinta, M., Giusti, E.: Nonlinear elliptic systems with quadratic growth. Manu. Math. 24, 323--349 (1978)zbMATHGoogle Scholar
  22. 22.
    Giaquinta, M., Giusti, E.: On the regularity of minima of variational integrals. Acta Math. 142, 31--46 (1982)MathSciNetGoogle Scholar
  23. 23.
    Gromov, M.: Carnot-Carthéodory spaces seen from within. Sub-Rieman. Geometry, Ed. A. Bellaiche, J. Risler, Progress in Mathematics 114 (1996), Birkhäuser, BaselGoogle Scholar
  24. 24.
    Grafakos, L., Rochberg, R.: Compensated compactness and the Heisenberg group. Math. Ann. 301, 601--611 (1995)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hélein, F.: Régularité des applications harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris 312, 591--596 (1991)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000)Google Scholar
  27. 27.
    Hildebrandt, S., Kaul, H., Widman, K.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138, 1--16 (1977)zbMATHGoogle Scholar
  28. 28.
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147--171 (1967)Google Scholar
  29. 29.
    Hamilton, R.: Harmonic maps of manifolds with boundary. Springer LNM 471 (1975)Google Scholar
  30. 30.
    Hajlasz, P., Strzelecki, P.: Subelliptic \(p\)-harmonic maps into spheres and the ghost of Hardy spaces. Math. Ann. 312, 341--362 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Martin, G., Iwaniec, T.: Quasiregular mappings in even dimensions. Acta Math. 170, 29--81 (1993)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Jerison, D.: Poincaré inequality for vector fields satisfying Hörmander's condition. Duke Math. J. 53, 503--523 (1986)MathSciNetzbMATHGoogle Scholar
  33. 33.
    John, F., Nirenberg, L.: On functions of bounded mean oscillations. CPAM. 14, 415--426 (1961)zbMATHGoogle Scholar
  34. 34.
    Jost, J., Xu, J.: Subelliptic harmonic maps. Trans. AMS, 350(11), 4633--4649 (1998)Google Scholar
  35. 35.
    Lu, G.: The sharp Poincaré inequality for free vector fields: An endpoint result. Rev. Mat. Ibe. 10, 453--466 (1994)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Müller, S.: Higher integrability of determinants and weak convergence in \(L^1\). J. Reine Angew. Math. 412, 20--34 (1990)Google Scholar
  37. 37.
    Nagel, A., Stein, E., Wainger, S.: Balls and metrics defined by vector fields I: basic properties. Acta Math. 155, 103--147 (1985)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Price, P.: A monotonicity formula for Yang-Mills fields. Manu. Math. 43, 131--166 (1983)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Riviére, T.: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175, 197--226 (1995)MathSciNetGoogle Scholar
  40. 40.
    Rothschild, L., Stein, E.: Hypoelliptic operators and nilpotent Lie groups. Acta Math. 137, 247--320 (1977)zbMATHGoogle Scholar
  41. 41.
    Sánchez-Calle, A.: Fundamental solutions and geometry of the sums of squares of vector fields. Invent. Math. 78, 143--160 (1984)Google Scholar
  42. 42.
    Semmes, S.: A primer on Hardy spaces and some remarks on a theorem of Evans and Müller. Comm. PDE 19, 277--319 (1994)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Diff. Geom. 17, 307--335 (1982)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Varopoulos, T., Saloff-Coste, L., Coulhon, T.: Analysis and geometry on groups. Cambridge Univ. Press, 1992Google Scholar
  45. 45.
    Varadarajan, V.: Lie groups, Lie algebras and Their Representation. Prentice Hall, New York, 1974Google Scholar
  46. 46.
    Xu, J.: Regularity for quasilinear second order subelliptic equations. CPAM 45, 77--96 (1992)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Xu, J., Zuily, C.: Higher interior regularity for quasilinear subelliptic systems. Cal. Var. 5, 323--343 (1997)CrossRefzbMATHGoogle Scholar
  48. 48.
    Zhou, Z.: Uniqueness of subelliptic harmonic maps. Ann. Global Anal. Geom. 17, 581--594 (1999)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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