1,721,036 research outputs found
On the existence of closed biconservative surfaces in space forms
Full text linkBiconservative surfaces of Riemannian 3-space forms N3(ρ), are either constant mean curvature (CMC) surfaces or rotational lin- ear Weingarten surfaces verifying the relation 3κ1 + κ2 = 0 be- tween their principal curvatures κ1 and κ2. We characterise the profile curves of the non-CMC biconservative surfaces as the crit- ical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC bicon- servative surfaces in the round 3-sphere, S3(ρ). However, none of these closed surfaces is embedded in S3(ρ)
Triharmonic Curves in 3-Dimensional Homogeneous Spaces
Full text linkWe first prove that, unlike the biharmonic case, there exist tri- harmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classifica- tion of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces
Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces
No full textA hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi-Cartan-Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only non-trivial example of a totally biharmonic surface appears in the product space S^2(ho)xR
Subelliptic biharmonic maps
No full textWe study subelliptic biharmonic maps i.e. smooth maps from a compact strictly pseudoconvex CR manifold
into a Riemannian manifold which are critical points of the energy functional . We show that is a
sublelliptic biharmonic map if and only if its vertical lift to the (total space of the) canonical
circle bundle is a biharmonic map with respect to the Fefferman metric on
Polyharmonic Helices
Full text linkThe main aim of this paper is to investigate the existence of
Frenet helices which are polyharmonic of order r, shortly, r-harmonic.
We shall obtain existence, non-existence and classification results. More specifically, we obtain a complete classification of proper r-harmonic helices into the 3-dimensional solvable Lie group Sol_3. Next, we investigate the existence of proper r-harmonic helices into Bianchi–Cartan–Vranceanu spaces and, in this context, we find new examples. Finally, we shall establish some non-existence results both for Frenet curves and Frenet helices of order n ≥ 4 when the ambient space is the Euclidean sphere S^m
Index and nullity of proper biharmonic maps in spheres
No full textIn recent years, the study of the bienergy functional has attracted the attention of a large community of researchers, but there are not many examples where the second variation of this functional has been thoroughly studied. We shall focus on this problem and, in particular, we shall compute the exact index and nullity of some known examples of proper biharmonic maps. Moreover, we shall analyze a case where the domain is not compact. More precisely, we shall prove that a large family of proper biharmonic maps φ: R->S^2 is strictly stable with respect to compactly supported variations. In general, the computations involved in this type of problems are very long. For this reason, we shall also define and apply to specific examples a suitable notion of index and nullity with respect to equivariant variations
On the second variation of the biharmonic Clifford torus in S-4
Full text linkThe flat torus T = S-1 (1/2) x S-1 (1/2) admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere S-4 given by Phi = i o phi, where phi : T -> S-3 (1/root 2) is the minimal Clifford torus and i : S-3 (1 root 2) -> S-4 is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion Phi. After, we shall study in the detail the kernel of the generalised Jacobi operator I-2 Phi. We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper, we shall analyse the specific contribution of phi to the biharmonic index and nullity of Phi. In this context, we shall study a more general composition (Phi) over tilde = i o (phi) over tilde, where (phi) over tilde : M-m -> Sn-1 (1/root 2), m >= 1, n >= 3, is a minimal immersion and i : Sn-1 (1/root 2) -> S-n is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of (Phi) over tilde is nonnegatively defined on C((phi) over tilde -1TSn-1(1/root 2)). Then, we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the p-harmonic index and nullity of phi. In the final section, we compare our general results with those which can be deduced from the study of the equivariant second variation
New examples of r-harmonic immersions into the sphere
No full textPolyharmonic, or r-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by EellsâLemaire in 1983. The main aim of this paper is to construct new examples of proper r-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion i:Snâ1(R)âaSnis a proper r-harmonic submanifold of Snif and only if the radius R is equal to 1/r. We shall also prove the existence of proper r-harmonic generalized Clifford's tori into the sphere
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