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Triharmonic isometric immersions into a manifold of non-positively constant curvature

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Abstract

A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study the generalized Chen’s conjecture for a triharmonic isometric immersion \(\varphi \) into a space form of non-positive constant curvature. We show that if the domain is complete and both the 4-energy of \(\varphi \), and the \(L^4\)-norm of the tension field \(\tau (\varphi )\), are finite, then such an immersion \(\varphi \) is minimal.

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Authors and Affiliations

  1. Division of Mathematics, Shimane University, Nishikawatsu 1060, Matsue, 690-8504, Japan

    Shun Maeta

  2. Graduate School of Science and Engineering, Yamaguchi University, Yamaguchi, 753-8512, Japan

    Nobumitsu Nakauchi

  3. Institute for International Education, Tohoku University, Kawauchi 41, Sendai, 980-8576, Japan

    Hajime Urakawa

Authors
  1. Shun Maeta
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  2. Nobumitsu Nakauchi
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  3. Hajime Urakawa
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Correspondence to Shun Maeta.

Additional information

Communicated by A. Cap.

S. Maeta supported by the Grant-in-Aid for Research Activity Start-up, Japan Society for the Promotion of Science, No. 25887044.

N. Nakauchi supported by the Grant-in-Aid for the Scientific Research (C), Japan Society for the Promotion of Science, No 24540213.

H. Urakawa supported by the Grant-in-Aid for the Scientific Research (C), Japan Society for the Promotion of Science, No 25400154.

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Maeta, S., Nakauchi, N. & Urakawa, H. Triharmonic isometric immersions into a manifold of non-positively constant curvature. Monatsh Math 177, 551–567 (2015). https://doi.org/10.1007/s00605-014-0713-4

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  • DOI: https://doi.org/10.1007/s00605-014-0713-4

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