Browse > Article
http://dx.doi.org/10.4134/CKMS.c210086

ON THE CONFORMAL TRIHARMONIC MAPS  

Ouakkas, Seddik (Laboratory of Geometry, Analysis, Control and Applications University of Saida, Dr Moulay Tahar)
Reguig, Yasmina (Laboratory of Geometry, Analysis, Control and Applications University of Saida, Dr Moulay Tahar)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.2, 2022 , pp. 607-629 More about this Journal
Abstract
In this paper, we give the necessary and sufficient condition for the conformal mapping ϕ : (ℝn, g0) → (Nn, h) (n ≥ 3) to be triharmonic where we prove that the gradient of its dilation is a solution of a fourth-order elliptic partial differential equation. We construct some examples of triharmonic maps which are not biharmonic and we calculate the trace of the stress-energy tensor associated with the triharmonic maps.
Keywords
Conformal map; harmonic map; biharmonic map; triharmonic map;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 V. Branding, The stress-energy tensor for polyharmonic maps, Nonlinear Anal. 190 (2020), 111616, 17 pp. https://doi.org/10.1016/j.na.2019.111616   DOI
2 N. Nakauchi and H. Urakawa, Polyharmonic maps into the Euclidean space, Note Mat. 38 (2018), no. 1, 89-100. https://doi.org/10.5840/ancientphil20183815   DOI
3 P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, 29, The Clarendon Press, Oxford University Press, Oxford, 2003. https://doi.org/10.1093/acprof:oso/9780198503620.001.0001   DOI
4 P. Goldstein, P. Strzelecki, and A. Zatorska-Goldstein, On polyharmonic maps into spheres in the critical dimension, Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009), no. 4, 1387-1405. https://doi.org/10.1016/j.anihpc.2008.10.008   DOI
5 S. Maeta, k-harmonic maps into a Riemannian manifold with constant sectional curvature, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1835-1847. https://doi.org/10.1090/S0002-9939-2011-11049-9   DOI
6 S. Maeta, Construction of triharmonic maps, Houston J. Math. 41 (2015), no. 2, 433-444.
7 S. Ouakkas and D. Djebbouri, Conformal maps, biharmonic maps, and the warped product, Mathematics 15 (2016), no. 4. https://doi.org/10.3390/math4010015   DOI
8 V. Branding, A structure theorem for polyharmonic maps between Riemannian manifolds, J. Differential Equations 273 (2021), 14-39. https://doi.org/10.1016/j.jde.2020.11.046   DOI
9 S. Maeta, The second variational formula of the k-energy and k-harmonic curves, Osaka J. Math. 49 (2012), no. 4, 1035-1063. http://projecteuclid.org/euclid.ojm/1355926886
10 P. Baird, A. Fardoun, and S. Ouakkas, Conformal and semi-conformal biharmonic maps, Ann. Global Anal. Geom. 34 (2008), no. 4, 403-414. https://doi.org/10.1007/s10455-008-9118-8   DOI
11 G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389-402.
12 E. Loubeau and Y.-L. Ou, Biharmonic maps and morphisms from conformal mappings, Tohoku Math. J. (2) 62 (2010), no. 1, 55-73. https://doi.org/10.2748/tmj/1270041027   DOI