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Stefano Montaldo

Other affiliations: University of Leeds
Bio: Stefano Montaldo is an academic researcher from University of Cagliari. The author has contributed to research in topics: Biharmonic equation & Mean curvature. The author has an hindex of 25, co-authored 108 publications receiving 2314 citations. Previous affiliations of Stefano Montaldo include University of Leeds.


Papers
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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.

249 citations

Journal ArticleDOI
TL;DR: In this article, the authors explicitly classify the nonharmonic biharmonic submanifolds of the unit three-dimensional sphere (3D) into three classes: (1).
Abstract: We explicitly classify the nonharmonic biharmonic submanifolds of the unit three-dimensional sphere ${\mathbb S}^3$.

204 citations

Journal Article
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

01 Jan 2002
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].

156 citations

Journal ArticleDOI
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.

121 citations


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Book
01 Jan 1970

329 citations

Journal ArticleDOI
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.

249 citations

Journal Article
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