1,721,058 research outputs found
Higher order energy functionals and the chen-maeta conjecture
Full text linkThe study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire 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 ErES (φ) = (1/2) ∫M|(d∗ + d)r (φ)|2 dV, where r ≥ 2 and φ: M → N is a map between two Riemannian manifolds. The initial part of this paper is a short overview on basic definitions, properties, recent developments and open problems concerning the functionals ErES (φ) and other, equally interesting, higher order energy functionals Er(φ) which were introduced and studied in various papers by Maeta and other authors. If a critical point φ of ErES (φ) (respectively, Er(φ)) is an isometric immersion, then we say that its image is an ES − r-harmonic (respectively, r-harmonic) submanifold of N. We observe that minimal submanifolds are trivially both ES − r-harmonic and r-harmonic. Therefore, it is natural to say that an ES − r-harmonic (r-harmonic) submanifold is proper if it is not minimal. In the special case that the ambient space N is the Euclidean space Rn the notions of ES − r-harmonic and r-harmonic submanifolds coincide. The Chen-Maeta conjecture is still open: it states that, for all r ≥ 2, any proper, r-harmonic submanifold of Rn is minimal. In the second part of this paper we shall focus on the study of G = SO(p + 1) × SO(q + 1)-invariant submanifolds of Rn, n = p + q + 2. In particular, we shall obtain an explicit description of the relevant Euler-Lagrange equations in the case that r = 3 and we shall discuss difficulties and possible developments towards the proof of the Chen-Maeta conjecture for 3-harmonic G-invariant hypersurfaces
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
On the Omori-Yau maximum principle and its applications to differential equations and geometry.
No full textAbstractIn this paper we prove a generalised version of the Omori-Yau maximum principle and describe some applications to problems in geometry and differential equations
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
Studio delle modificazioni dento-alveolari verticali indotte dal Lip Bumper: ulteriore contributo
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Studio delle modificazioni dento basali verticali indotte dal Lip-Bumper: contributo clinico
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