2-harmonic maps and their first and second variational formulas

By Guoying Jiang. 2008

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BIMINIMAL IMMERSIONS (2004, unknown): “The Euler–Lagrange operator attached to biharmonicity, called the bitension field and computed by Jiang [9], is τ2(φ) = −(∆τ(φ) − trR (dφ, τ(φ)) dφ), and vanishes if and only if the map φ is biharmonic.”
Biharmonic hypersurfaces in Riemannian manifolds (2009, background): “…umbilical biharmonic hypersurfaces in an Einstein space, and con- struct 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 Rm by parallel hyperplanes into a foliation with…”
Biharmonic maps and morphisms from conformal mappings (2008, background): “tively curved targets, the converse holds ([ 14 ]).”
Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature (2011, unknown): “A C map φ : (M, g) → (N, h) is called to be biharmonic ([2], [5], [7]) if τ2(φ) = 0.”