Author

# E Loubeau

Bio: E Loubeau is an academic researcher from University of Cagliari. The author has contributed to research in topics: Tensor & Biharmonic equation. The author has an hindex of 1, co-authored 1 publications receiving 145 citations.

Topics: Tensor, Biharmonic equation, Harmonic map

##### Papers

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TL;DR: In this paper, the geometric properties of biharmonic curves and surfaces of some Thurston's geometries have been discussed, including the biharmonicity of maps between warped products.

Abstract: points of the bienergy functional E2(’) = 1 R M j?(’)j 2 vg; where ?(’) is the tension fleld of ’. Biharmonic maps are a natural expansion of harmonic maps (?(’) = 0). Although E2 has been on the mathematical scene since the early ’60, when some of its analytical aspects have been discussed, and regularity of its critical points is nowadays a well-developed fleld, a systematic study of the geometry of biharmonic maps has started only recently. In this lecture we focus on the geometric properties of biharmonic maps and describe some recent achievements on the subject: (a) We give the explicit classiflcations of biharmonic curves and surfaces of some Thurston’s geometries [2, 3, 4]. (b) We describe the biharmonicity of maps between warped products and using this setting we study three classes of axially symmetric biharmonic maps [1]. (c) Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy, show it derives from a variational problem on metrics, exhibit the peculiarity of dimension four, and use the stress-energy tensor to construct new examples of biharmonic maps [5].

145 citations

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TL;DR: In this paper, the authors give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphere, where the bi-harmonic equation is solved explicitly.

Abstract: We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereS
n
. In the case of curves inS
n
we solve explicitly the biharmonic equation.

233 citations

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TL;DR: In this paper, a natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively.

Abstract: and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in C∞(M,N) if and only if it is a minimal immersion [26]. Thus, in order to study minimal immersions one can look at harmonic Riemannian immersions. A natural generalization of harmonic maps and minimal immersions can be given by considering the functionals obtained integrating the square of the norm of the tension field or of the mean curvature vector field, respectively. More precisely: • biharmonic maps are the critical points of the bienergy functional E2 : C∞(M,N) → R, E2(φ) = 12 ∫

178 citations

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TL;DR: In this article, the generalized Chen conjecture is proven to be true for totally umbilical biharmonic hypersurfaces in an Einstein space, and a 2-parameter family of conformally flat metrics and a 4-parameters family of multiply warped product metrics, each of which turns the foliation of an upper-half space of a Riemannian manifold by parallel hyperplanes into a foliation with each leaf a proper hypersurface.

Abstract: We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied by Jiang, Chen, Caddeo, Montaldo, and Oniciuc. We then apply the equation to show that the generalized Chen conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a 2-parameter family of conformally flat metrics and a 4-parameter family of multiply warped product metrics, each of which turns the foliation of an upper-half space of R m by parallel hyperplanes into a foliation with each leaf a proper biharmonic hypersurface. We also study the biharmonicity of Hopf cylinders of a Riemannian submersion.

134 citations

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TL;DR: In this article, Chen et al. studied the rigidity of pseudoumbilical biharmonic submanifolds of codimension 2 and for B-Y surfaces with parallel mean curvature vector field.

Abstract: We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.

113 citations