Polyharmonic hypersurfaces into space forms

Montaldo, Stefano; Oniciuc, Cezar; Ratto, Andrea

doi:10.1007/s11856-022-2315-5

Cited by 17 publications

(26 citation statements)

References 48 publications

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“…Things drastically change when the ambient is the Euclidean sphere m+1 . Indeed, in this case several examples of proper r-harmonic hypersurfaces have been constructed and studied (see [6,22,26,27] and references therein).…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…In our recent work [26], we proved some general results for r-harmonic hypersurfaces into space forms and deduced that the value of the integer r plays a crucial role to generate geometric phenomena which differ substantially from the classical situation corresponding to the biharmonic and triharmonic cases. For instance, if ≥ 3 , there exists no isoparametric hypersurface of m+1 of degree which is proper biharmonic or triharmonic.…”

Section: Introductionmentioning

confidence: 85%

“…Next, we obtain some geometric results for triharmonic surfaces which provide a version in the pseudo-Riemannian case of some facts which we proved in [26] in the Riemannian case.…”

Section: Remark 314mentioning

confidence: 97%

“…Therefore, a reasonable starting point is to focus on the case that the ambient is a space form N m+1 (c) (here and below, c denotes the sectional curvature) and study CMC hypersurfaces with constant squared norm ‖A‖ 2 of the shape operator. In this order of ideas, in the Riemannian case we obtained the following general result: Theorem 1.1 [26] Let M m be a non-minimal CMC hypersurface in a Riemannian space form N m+1 (c) and assume that ‖A‖ 2 is constant. Then M m is proper r-harmonic ( r ≥ 3 ) if and only if where the constant denotes the mean curvature of M m .…”

Section: Introductionmentioning

confidence: 99%

“…For instance, if ≥ 3 , there exists no isoparametric hypersurface of m+1 of degree which is proper biharmonic or triharmonic. By contrast, when r ≥ 5 , there are several examples of such hypersurfaces which are proper r-harmonic (see [26]). From the point of view of the differential geometry of submanifolds, the difficulties which one encounters in studying the equations which define a general r-harmonic submanifold are huge.…”

Section: Introductionmentioning

confidence: 99%

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Polyharmonic hypersurfaces into pseudo-Riemannian space forms

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Montaldo

Oniciuc

et al. 2022

Annali di Matematica

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In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form $$N^{m+1}_t(c)$$ N t m + 1 ( c ) of dimension $$m+1$$ m + 1 and index t ($$m\ge 2$$ m ≥ 2 and $$1 \le t\le m$$ 1 ≤ t ≤ m ). We shall study hypersurfaces $${M}^{m}_{t'}$$ M t ′ m which are polyharmonic of order r (briefly, r-harmonic), where $$r\ge 3$$ r ≥ 3 and either $$t'=t$$ t ′ = t or $$t'=t-1$$ t ′ = t - 1 . Let A denote the shape operator of $${M}^{m}_{t'}$$ M t ′ m . Under the assumptions that $${M}^{m}_{t'}$$ M t ′ m is CMC and $${{\rm{Tr}}}\,{A}^{2}$$ Tr A 2 is a constant, we shall obtain the general condition which determines that $${M}^{m}_{t'}$$ M t ′ m is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces ($$r \ge 3$$ r ≥ 3 ). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.

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Section: Statement Of the Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 85%

“…Next, we obtain some geometric results for triharmonic surfaces which provide a version in the pseudo-Riemannian case of some facts which we proved in [26] in the Riemannian case.…”

Section: Remark 314mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations