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Liouville Theorems for F-Harmonic Maps and Their Applications

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Abstract

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the maps at infinity. In particular, the results can be applied to F-harmonic maps from some pinched manifolds, and can deduce a Bernstein type result for an entire minimal graph.

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References

  1. Ara M.: Geometry of F-harmonic maps. Kodai Math. 22, 243–263 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronszajn N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. 36, 235–249 (1957)

    MathSciNet  MATH  Google Scholar 

  3. Baird P.: Stress-energy tensors and the Lichnerowicz Laplacian. J. Geom. Phys. 58, 1329–1342 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheng, S.Y.: Liouville theorem for harmonic maps. Pure Math., vol. 36, pp. 147–151. American Mathematical Society, Providence (1980)

  5. Dong Y.X., Wei S.W.: On vanishing theorems for vector bundle valued p-forms and their applications. Commun. Math. Phys. 304, 329–368 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Garber W.D., Ruijsennaars S.N.M., Seiler E., Burns D.: On finite action solutions of the nonlinear σ-model. Ann. Phys. 119, 305–325 (1979)

    Article  MathSciNet  Google Scholar 

  7. Greene, R.E., Wu, H.: Function theory on manifolds which posses a pole, Lecture Notes in Mathematics, vol. 699. Springer, Berlin (1979)

  8. Hildebrandt S.: Liouville theorems for harmonic mappings, and an approach to Bernstein theorems. Ann. Math. Stud. 102, 107–131 (1982)

    MathSciNet  MATH  Google Scholar 

  9. Jin Z.R.: Liouville theorems for harmonic maps. Invent. Math. 108, 1–10 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kassi M.: A Liouville theorem for F-harmonic maps with finite F-energy. Electron. J. Differ. Equ. 15, 1–9 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Kazdan J.L.: Unique continuation in geometry. Commun. Pure Appl. Math. 41(5), 667–681 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Otway T.H.: Maps and fields with compressible density. Rend. Sem. Mat. Univ. Padova 111, 133–159 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Otway T.H.: Geometric analysis near and across a sonic curve. In: Bendo, C.V. (eds) New Developments in Mathematical Physics Research, pp. 27–45. Nova Science Publisher, New York (2004)

    Google Scholar 

  14. Pigola, S., Rigoli, M., Setti, A.G.: Vanishing and finiteness results in geometric analysis: a generalization of the Bochner technique. Progress in Mathematics, vol. 266 (2008)

  15. Sealey H.C.J.: Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang–Mills theory. Math. Proc. Camb. Philos. Soc. 91, 441–452 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Simons J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88(2), 62–105 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  17. Simon L.: Asymptotic behaviour of minimal graphs over exterior domains. Ann. Inst. Henri Poincaŕe 4(3), 231–242 (1987)

    MathSciNet  MATH  Google Scholar 

  18. Schoen R., Yau S.T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Commun. Math. Helv. 51, 333–341 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  19. Xin Y.L.: Differential forms, conservation law and monotonicity formula. Sci. Sin. (Ser. A) XXIX, 40–50 (1986)

    MathSciNet  MATH  Google Scholar 

  20. Yang Y.S.: Classical solutions in the Born–Infeld theory. Proc. R. Soc. Lond. A 456, 615–640 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences and Laboratory of Mathematics, for Nonlinear Science, Fudan University, Shanghai, 200433, China

    Yuxin Dong

  2. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350108, China

    Hezi Lin

  3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Guilin Yang

Authors
  1. Yuxin Dong
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  2. Hezi Lin
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  3. Guilin Yang
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Corresponding author

Correspondence to Hezi Lin.

Additional information

Y. Dong was supported by NSFC Grant No 11271071. and H. Lin was supported by NSFC Grant No 11401099.

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Dong, Y., Lin, H. & Yang, G. Liouville Theorems for F-Harmonic Maps and Their Applications. Results. Math. 69, 105–127 (2016). https://doi.org/10.1007/s00025-015-0480-0

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  • DOI: https://doi.org/10.1007/s00025-015-0480-0

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