Non-existence of quadratic harmonic maps of 𝑆⁴ into 𝑆⁵ or 𝑆⁶

Wu, Faen; Zhao, Xinnuan

doi:10.1090/S0002-9939-2012-11460-1

Non-existence of quadratic harmonic maps of $S^{4}$ into $S^{5}$ or $S^{6}$
HTML articles powered by AMS MathViewer

by Faen Wu and Xinnuan Zhao PDF

Proc. Amer. Math. Soc. 141 (2013), 1083-1091 Request permission

Abstract:

In this paper, we settle the last two open cases of non-existence of full quadratic harmonic maps from $S^{4}$ to $S^{5}$ or $S^{6}$. Assume that there exist full quadratic harmonic maps from $S^{4}$ to $S^{n}$ for some integer $n$. As a consequence of our theorem we obtain that the sufficient and necessary condition of the existence of such maps is that $n$ satisfy $4\leq n\leq 13$ and $n\neq 5,6$.

References

J. W. S. Cassels, On the representation of rational functions as sums of squares, Acta Arith. 9 (1964), 79–82. MR 162791, DOI 10.4064/aa-9-1-79-82
James Eells and Luc Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 703510, DOI 10.1090/cbms/050
James Eells and Andrea Ratto, Harmonic maps and minimal immersions with symmetries, Annals of Mathematics Studies, vol. 130, Princeton University Press, Princeton, NJ, 1993. Methods of ordinary differential equations applied to elliptic variational problems. MR 1242555, DOI 10.1515/9781400882502
Hillel Gauchman and Gábor Tóth, Constructions of harmonic polynomial maps between spheres, Geom. Dedicata 50 (1994), no. 1, 57–79. MR 1280796, DOI 10.1007/BF01263652
Gábor Tóth, Harmonic maps and minimal immersions through representation theory, Perspectives in Mathematics, vol. 12, Academic Press, Inc., Boston, MA, 1990. MR 1042100
Gábor Tóth, Classification of quadratic harmonic maps of $S^3$ into spheres, Indiana Univ. Math. J. 36 (1987), no. 2, 231–239. MR 891772, DOI 10.1512/iumj.1987.36.36013
Huixia He, Hui Ma, and Feng Xu, On eigenmaps between spheres, Bull. London Math. Soc. 35 (2003), no. 3, 344–354. MR 1960944, DOI 10.1112/S002460930200190X
Kee Yuen Lam, Some new results on composition of quadratic forms, Invent. Math. 79 (1985), no. 3, 467–474. MR 782229, DOI 10.1007/BF01388517
Zizhou Tang, Harmonic polynomial morphisms between Euclidean spaces, Sci. China Ser. A 42 (1999), no. 6, 570–576. MR 1717002, DOI 10.1007/BF02880074
Zizhou Tang, New constructions of eigenmaps between spheres, Internat. J. Math. 12 (2001), no. 3, 277–288. MR 1841516, DOI 10.1142/S0129167X01000812
Keisuke Ueno, Some new examples of eigenmaps from $S^m$ into $S^n$, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 6, 205–208. MR 1232826
Paul Y. H. Yiu, Quadratic forms between spheres and the nonexistence of sums of squares formulae, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 493–504. MR 857724, DOI 10.1017/S0305004100066226