A short survey on biharmonic maps between Riemannian manifolds

By Stefano Montaldo, C. Oniciuc. 2005

197 citations recorded by the source corpus.

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 ∫

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

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Citations in the corpus

The stress-energy tensor for biharmonic maps (2006, background): “See [17] for an account of biharmonic maps and (The Bibliography of Biharmonic Maps.”
Biharmonic maps into a Riemannian manifold of non-positive curvature (2012, background): “Jiang [15] studied the first and second variation formulas of E2, extensive studies in this area have been done (for instance, see [2], [4], [18], [19], [21], [24], [26], [12], [13], [14], etc.”
Biharmonic submanifolds of $${\mathbb{C}P^n}$$ (2009, background): “The analytic aspects of biharmonic maps as well as the differential geometry of such maps have been studied in the last decade (see, for example, [7, 17, 22, 23] and [2, 10, 16, 21, 24], respectively).”
Biharmonic hypersurfaces in Riemannian manifolds (2009, background): “Biharmonic submanifolds in other model spaces: For the study of biharmonic curves in various model spaces we refer the readers to the survey article [18], and for special biharmonic submanifolds in contact manifolds or Sasakian space forms see recent works [1], [13], [10], [11], [24], and [25].”