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Exponentially harmonic maps, Morse index and Liouville type theorems

Abstract

We obtain a result on the Morse index of an exponentially harmonic map from a Riemannian manifold into the unit n-sphere. Next, we prove a Liouville type 1 theorem for exponentially harmonic maps between two Riemannian manifolds. Finally, let \((M, g_0)\) be a complete Riemannian manifold with a pole \(x_0\) and (Nh) a Riemannian manifold, under certain conditions we establish a Liouville type 2 theorem for exponentially harmonic maps \(f:(M, \rho ^2 g_0)\rightarrow N\), \(0< \rho \in C^\infty (M)\).

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Correspondence to Yuan-Jen Chiang.

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Chiang, YJ. Exponentially harmonic maps, Morse index and Liouville type theorems. European Journal of Mathematics 6, 1388–1402 (2020). https://doi.org/10.1007/s40879-019-00362-3

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  • DOI: https://doi.org/10.1007/s40879-019-00362-3

Keywords

  • Exponentially harmonic map
  • Morse index
  • Liouville type theorems

Mathematics Subject Classification

  • 58E20
  • 58G11
  • 35J20