Geodesic
Deformations of Metrics and Biharmonic Maps
Communications in Mathematics, Tome 28 (2020) no. 3, pp. 263-275.

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We construct biharmonic non-harmonic maps between Riemannian manifolds $(M,g)$ and $(N,h)$ by first making the ansatz that $\varphi \colon (M,g) \rightarrow (N,h)$ be a harmonic map and then deforming the metric on $N$ by $$\tilde {h}_{\alpha }=\alpha h+(1-\alpha )df\otimes df$$ to render $\varphi $ biharmonic, where $f$ is a smooth function with gradient of constant norm on $(N,h)$ and $\alpha \in (0,1)$. We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.
Classification : 53C20, 53C22, 58E20
Keywords: Riemannian geometry; Harmonic maps; Biharmonic maps 
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     title = {Deformations of {Metrics} and {Biharmonic} {Maps}},
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Benkartab, Aicha; Cherif, Ahmed Mohammed. Deformations of Metrics and Biharmonic Maps. Communications in Mathematics, Tome 28 (2020) no. 3, pp. 263-275. http://geodesic.mathdoc.fr/item/COMIM_2020__28_3_a1/