On the regularity theories for harmonic maps from Finsler manifolds

Abstract

The author studies the regularity of energy minimizing maps from Finsler manifolds to Riemannian manifolds. It is also shown that the energy minimizing maps are smooth, when the target manifolds have no focal points.

Keywords

Finsler manifolds / Harmonic map / Regularity

References

[1] Morrey C. B.. The problem of Plateau on a Riemannian manifold. Ann. of Math., 1948, 49(4): 807-851

[2] Hilderbrandt S., Kaul H., Widman K. O.. An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math., 1977, 138(1): 1-16

[3] Lemaire L.. Applications harmoniques de surfaces Riemanniennes. J. Diff. Geom., 1978, 13(1): 51-78

[4] Schoen R., Uhlenbeck K.. A regularity theory for harmonic maps. J. Diff. Geom., 1982, 17(2): 307-335

[5] Schoen R., Uhlenbeck K.. Boundary regularity and the Dirichlet problem for harmonic maps. J. Diff. Geom., 1983, 18(2): 253-268

[6] Mo X. H.. Harmonic maps from Finsler manifolds. Illinois J. Math., 2001, 45(4): 1331-1345

[7] Shen Y. B., Zhang Y.. Second variation of harmonic maps between Finsler manifolds. Sci. China Ser. A, 2004, 47(1): 39-51

[8] He Q., Shen Y. B.. Some results on harmonic maps for Finsler manifolds. Inter. J. Math., 2005, 16(9): 1017-1031

[9] He Q., Shen Y. B.. Some properties of harmonic maps between Finsler manifolds. Houston J. Math., 2007, 33(3): 683-699

[10] Shen Y. B., Wei S. W.. The stability of harmonic maps on Finsler manifolds. Houston J. Math., 2008, 34(4): 1049-1056

[11] Mo X. H., Yang Y.. The existence of harmonic maps from Finsler manifolds to Riemannian manifolds. Sci. China Ser. A, 2005, 48(1): 115-130

[12] von der Mosel H., Winklmann S.. On weakly harmonic maps from Finsler to Riemannian manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2009, 26(1): 39-57

[13] Mo X. H., Zhao L.. Regularity of weakly harmonic maps from a Finsler surface into an n-sphere. Pacific J. Math., 2011, 253(1): 145-155

[14] Xin Y. L.. Some existence theorems for harmonic maps in Riemannian manifolds. Sci. Sinica, 1982, 25(12): 1271-1278