The stress-energy tensor for biharmonic maps

By Eric Loubeau, Stefano Montaldo, C. Oniciuc. 2006

76 citations recorded by the source corpus.

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Biomedical Engineering

View Paper: https://www.semanticscholar.org/paper/bd4ac6885dd952b562f063baac08c070fecc7dab

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In Applied Mathematics

1. Stefano Montaldo
2. C. Oniciuc
3. A. Balmuş
4. A. Ratto
5. Ye-lin Ou
6. Eric Loubeau
7. Bang‐Yen Chen

In Astronomy and Astrophysics

1. Stefano Montaldo
2. C. Oniciuc
3. A. Balmuş
4. A. Ratto
5. Ye-lin Ou
6. Eric Loubeau

Citations in the corpus

• Proper biconservative immersions into the Euclidean space (2013, background): “The study of the stress-energy tensor for the bienergy was initiated in [21] and afterwards developed in [22].”
• Biharmonic PNMC submanifolds in spheres (2011, background): “We note that the tangent part of τ2(φ) vanishes if and only if the stress-energy tensor for biharmonic maps associated to φ vanishes (see [22, 18]).”