Deformations of Metrics and Biharmonic Maps

Aicha Benkartab; Ahmed Mohammed Cherif

Communications in Mathematics (2020)

  • Volume: 28, Issue: 3, page 263-275
  • ISSN: 1804-1388

Abstract

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We construct biharmonic non-harmonic maps between Riemannian manifolds ( M , g ) and ( N , h ) by first making the ansatz that ϕ : ( M , g ) → ( N , h ) be a harmonic map and then deforming the metric on N by h ˜ α = α h + ( 1 - α ) d f ⊗ d f to render ϕ biharmonic, where f is a smooth function with gradient of constant norm on ( N , h ) and α ∈ ( 0 , 1 ) . We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

How to cite

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Benkartab, Aicha, and Cherif, Ahmed Mohammed. "Deformations of Metrics and Biharmonic Maps." Communications in Mathematics 28.3 (2020): 263-275. <http://eudml.org/doc/297226>.

@article{Benkartab2020,
abstract = {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.},
author = {Benkartab, Aicha, Cherif, Ahmed Mohammed},
journal = {Communications in Mathematics},
keywords = {Riemannian geometry; Harmonic maps; Biharmonic maps},
language = {eng},
number = {3},
pages = {263-275},
publisher = {University of Ostrava},
title = {Deformations of Metrics and Biharmonic Maps},
url = {http://eudml.org/doc/297226},
volume = {28},
year = {2020},
}

TY - JOUR
AU - Benkartab, Aicha
AU - Cherif, Ahmed Mohammed
TI - Deformations of Metrics and Biharmonic Maps
JO - Communications in Mathematics
PY - 2020
PB - University of Ostrava
VL - 28
IS - 3
SP - 263
EP - 275
AB - 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.
LA - eng
KW - Riemannian geometry; Harmonic maps; Biharmonic maps
UR - http://eudml.org/doc/297226
ER -

References

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