Abstract
In this paper we shall consider polyharmonic hypersurfaces of order r (briefly, r-harmonic hypersurfaces), where r ≥ 3 is an integer, into a space form Nm+1 (c) of curvature c. For this class of hypersurfaces we shall prove that, if c ≤ 0, then any r-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is \({\mathbb{S}^{m + 1}}\), we shall obtain the geometric condition which characterizes the r-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we shall establish the bounds for these two constants. In particular, we shall prove the existence of several new examples of proper r-harmonic isoparametric hypersurfaces in \({\mathbb{S}^{m + 1}}\) for suitable values of m and r. Finally, we shall show that all these r-harmonic hypersurfaces are also ES-r-harmonic, i.e., critical points of the Eells-Sampson r-energy functional.
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The authors would like to thank the referees for their valuable comments which have helped to improve the manuscript.
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S.M. and A.R. were supported by Fondazione di Sardegna (project STAGE) and Regione Autonoma della Sardegna (Project KASBA).
C.O. was partially supported by grant PN-III-P4-ID-PCE-2020-0794.
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Montaldo, S., Oniciuc, C. & Ratto, A. Polyharmonic hypersurfaces into space forms. Isr. J. Math. 249, 343–374 (2022). https://doi.org/10.1007/s11856-022-2315-5
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DOI: https://doi.org/10.1007/s11856-022-2315-5