On polyharmonic helices in space forms

Branding, Volker

doi:10.1007/s00013-022-01809-y

Cited by 4 publications

(2 citation statements)

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“…In the following, we will give a non-exhaustive overview on the current status of research on triharmonic maps. Triharmonic curves of constant curvature in various ambient geometries have been studied in [ 19 , 27 , 28 ], polyharmonic, and hence in particular triharmonic, helices in space forms were classified in [ 8 ]. The first attempt of explicitly constructing triharmonic maps in the sphere was carried out in [ 16 ] which was later systematically extended in [ 20 ].…”

Section: Introduction and Resultsmentioning

confidence: 99%

On the Normal Stability of Triharmonic Hypersurfaces in Space Forms

Branding

2023

J Geom Anal

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This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations. For the case of a spherical target we show that the normal index of the small proper triharmonic hypersphere $$\phi :\mathbb {S}^m(1/\sqrt{3})\hookrightarrow \mathbb {S}^{m+1}$$ ϕ : S m ( 1 / 3 ) ↪ S m + 1 is equal to one and make some comments on the normal stability of the proper triharmonic Clifford torus.

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Section: Introduction and Resultsmentioning

confidence: 99%

On the Normal Stability of Triharmonic Hypersurfaces in Space Forms

Branding

2023

J Geom Anal

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“…Note that in the case of dim M = 1 the functionals (1.4), (1.5) and (1.6) coincide and the critical points are given by polyharmonic curves. For the current status of research on this subject we refer to the recent article [9] and references cited in that manuscript. It can directly be seen that every harmonic map, that is a solution of τ (φ) = 0, automatically is a critical point of (1.4), (1.5) as well as of (1.6).…”

Section: Introduction and Resultsmentioning

confidence: 99%

On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps

Branding

2021

J Geom Anal

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Abstract4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first difference between these two variational problems.

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