Biharmonic Maps Between Riemannian Manifolds

By R. Caddeo, Eric Loubeau, Stefano Montaldo, C. Oniciuc, M. Piu. 2002

154 citations recorded by the source corpus.

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].

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/26804175b5af6f7113f5d2ad92638f5cd1be6fe9

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Citations in the corpus

Proper biconservative immersions into the Euclidean space (2013, methodology): “The decomposition of the bitension field with respect to its normal and tangent components was obtained with contributions of [3,7,23,25,26], and for hypersurfaces, it can be summarized in the following theorem.”
Biharmonic hypersurfaces in Riemannian manifolds (2009, background): “2.2. A constant mean curvature hypersurface in a Riemannian manifold is biharmonic if and only if it is minimal or, RicN(ξ,ξ) = |A|2 and (RicN(ξ))⊤ = 0. In particular, we recovered Proposition 2.4 in [21] which states that a constant mean curvature hypersurface in a Riemannian manifold (Nm+1,h) with”
Biharmonic PNMC submanifolds in spheres (2011, background): “ second fundamental form, H the mean curvature vector ﬁeld, f = |H| the mean curvature function, ∇⊥ and ∆⊥ the connection and the Laplacian in the normal bundle of M in N. This result was obtained in [10, 25] for submanifolds in space forms, and in [26] for general hypersurfaces. We note that the ta”
A short survey on biharmonic maps between Riemannian manifolds (2005, background): “s. If M is not compact, then the above argument can be used with the extra assumption that φis a Riemannian immersion and that the norm of τ(φ) is constant, as was shown by C. Oniciuc in Theorem 3.2 ([41]). Let φ: (M,g) → (N,h) be a Riemannian immersion. If |τ(φ)| is constant and RiemN ≤ 0, then φis”
Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature (2011, background): “t η be the mean curvature vector ﬁeld along ϕ deﬁned by η = 1 m Pm i=1B(ei,ei), where {ei}mi=1 is a local orthonormal frame on (M,g). Then, the generalizedB.Y.Chen’sconjecture(cf. [1], [2], [3], [4], [10], [11], [12]) is that: 2000 Mathematics Subject Classiﬁcation. 58E20. Key words and phrases. har”
The stress-energy tensor for biharmonic maps (2006, background): “10 For a vector field ζ on M , T M can be endowed with a Sasaki-type metric such that the projection π : T M → M is a Riemannian submersion and τ(π) = −(m + 1)ζ ◦ π ([18]).”
Explicit Formulas for Non-Geodesic Biharmonic Curves of the Heisenberg Group (2003, background): “e sectional curvature −1. • Other results about non-existence of non-harmonic biharmonic maps, when the sectional curvature or the Ricci curvature of M is non-positive, can be found in [Y1], [O1] and [O2]. It seems then natural, as the next step, to consider biharmonic submanifolds in a 3-manifold M”
Biharmonic hypersurfaces in 4‐dimensional space forms (2007, methodology): “The following characterization result, obtained in [7] and [18] by splitting the bitension field in its normal and tangent components, represents the main tool in the study of proper biharmonic submanifolds in space forms.”