Gradient Estimate and Liouville Theorems for p-Harmonic Maps

Abstract

In this paper, we first obtain an \(L^q\) gradient estimate for p-harmonic maps, by assuming the target manifold supporting a certain function, whose gradient and Hessian satisfy some analysis conditions. From this \(L^q\) gradient estimate, we get a corresponding Liouville type result for p-harmonic maps. Secondly, using these general results, we give various geometric applications to p-harmonic maps from complete manifolds with nonnegative Ricci curvature to manifolds with various upper bound on sectional curvature, under appropriate controlled images.

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Acknowledgements

The authors would also like to thank Professor G.F. Wang for his interest and suggestions. Yuxin Dong is supported by NSFC Grant No. 11771087. Hezi Lin is supported by NSFC Grant No. 11831005.

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Correspondence to Hezi Lin.

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Dong, Y., Lin, H. Gradient Estimate and Liouville Theorems for p-Harmonic Maps. J Geom Anal (2021). https://doi.org/10.1007/s12220-020-00594-w

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Keywords

  • p-Harmonic maps
  • Gradient estimate
  • Liouville theorems

Mathematics Subject Classification

  • Primary 53C21
  • 53C43
  • Secondary 35B53
  • 58E20