Proper<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>r</mml:mi></mml:math>-harmonic submanifolds into ellipsoids and rotation hypersurfaces

Montaldo, Stefano; Ratto, Andrea

doi:10.1016/j.na.2018.03.002

Cited by 17 publications

(9 citation statements)

References 21 publications

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“…Similarly, in our recent work [34] we proved the existence of several other Gequivariant examples of proper r-harmonic immersions and maps into rotation hypersurfaces and ellipsoids: again, it was not clearly stated that these examples were obtained by studying E r (ϕ) and not E ES r (ϕ). However, with the methods of the present paper, it is not difficult to verify that all the examples of [34] are not only r-harmonic, but also ES − r-harmonic. Therefore, we consider the present section of this work as a natural completion of [33] and [34].…”

Section: The Principle Of Symmetric Criticality and Existence Resultsmentioning

confidence: 79%

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Higher order energy functionals

Branding¹,

Montaldo²,

Oniciuc³

et al. 2019

Preprint

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The study of higher order energy functionals was first proposed by Sampson in 1965 and, later, by Eells andLemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called ES −r-energy functionals, where ϕ : M → N is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an ES − rharmonic map, i.e, a critical point of E ES r (ϕ). That seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of E ES r (ϕ) when N = S m (r ≥ 4, m ≥ 3), and we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for E ES r (ϕ) for r = 4. We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4, we shall also show that, even if 2r > dim M , the functionals E ES r (ϕ) may not satisfy the classical Palais-Smale Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.

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Section: The Principle Of Symmetric Criticality and Existence Resultsmentioning

confidence: 79%

“…However, with the methods of the present paper, it is not difficult to verify that all the examples of [34] are not only r-harmonic, but also ES − r-harmonic. Therefore, we consider the present section of this work as a natural completion of [33] and [34].…”

Section: The Principle Of Symmetric Criticality and Existence Resultsmentioning

confidence: 81%

Higher order energy functionals

Branding¹,

Montaldo²,

Oniciuc³

et al. 2019

Preprint

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“…Clearly, k-polyharmonic map equations are much more complicated than that of biharmonic maps, so few examples of k-polyharmonic maps have been found. For some recent work on k-polyharmonic maps see [21,22,11,12,13,14,17,18,16,19,6,7,8,9].…”

Section: Classification Of K-polyharmonic Conformal Maps Betweenmentioning

confidence: 99%

Some classifications of conformal biharmonic and k-polyharmonic maps

Ou¹

2021

Preprint

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We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a Möbius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of Möbius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.

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“…In this paper, we focus on the case k = 3, which is a particularly active subject recently (cf. [12,[14][15][16][17]25]). For instance, Maeta-Nakauchi-Urakawa [14] proved that a triharmonic isometric immersion into a Riemannian manifold of non-positive curvature must be minimal under certain suitable conditions.…”

Section: Introductionmentioning

confidence: 99%

Triharmonic CMC hypersurfaces in space forms with 4 distinct principal curvatures

Chen¹,

Guan²

2021

Preprint

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A triharmonic map is a critical point of the tri-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if M n (n ≥ 4) is a CMC proper triharmonic hypersurface in a space form R n+1 (c) with four distinct principal curvatures and the multiplicity of the zero principal curvature is at most one, then M has constant scalar curvature. In particular, we obtain any CMC proper triharmonic hypersurface in R 5 (c) is minimal when c ≤ 0, which supports the generalized Chen's conjecture. We also give some characterizations of CMC proper triharmonic hypersurfaces in S 5 .

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Properr-harmonic submanifolds into ellipsoids and rotation hypersurfaces

Cited by 17 publications

References 21 publications

Higher order energy functionals

Higher order energy functionals

Some classifications of conformal biharmonic and k-polyharmonic maps

Triharmonic CMC hypersurfaces in space forms with 4 distinct principal curvatures

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