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Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form

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Abstract

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in dimension at most 4. Techniques are given for determining when two solutions are linearly conformally inequivalent.

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References

  1. Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pac. J. Math. 195(1), 1–34 (2000)

    Article  MathSciNet  Google Scholar 

  2. Bryant, R.L.: Second order families of special Lagrangian 3-folds, Perspectives in Riemannian geometry. In: CRM Proceedings of Lecture Notes, vol. 40, Amer. Math. Soc., Providence, pp. 63–98 (2006)

  3. Cartan, E.: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45, 335–367 (1939)

    Article  MathSciNet  Google Scholar 

  4. Cartan, E.: Sur quelques familles remarquables d’hypersurfaces., C. R. Congr. Sci. Math. 30-41, (1939)

  5. Cecil, T.E.: Isoparametric and Dupin hypersurfaces. SIGMA Symm. Integ. Geom. Methods Appl. 4, 28 (2008). Paper 062

    MathSciNet  MATH  Google Scholar 

  6. Chen, Y., Kontsevich, M., Schwarz, A.: Symmetries of WDVV equations. Nuclear Phys. B 730(3), 352–363 (2005)

    Article  MathSciNet  Google Scholar 

  7. M. P. do Carmo and N. R. Wallach, : Minimal immersions of spheres into spheres. Ann. Math. (2) 93, 43–62 (1971)

  8. Dong, C., Griess Jr., R.L.: Rank one lattice type vertex operator algebras and their automorphism groups. J. Algebra 208(1), 262–275 (1998)

    Article  MathSciNet  Google Scholar 

  9. Fickus, M., Jasper, J., Mixon, D.G., Peterson, J.D., Watson, C.E.: Equiangular tight frames with centroidal symmetry. Appl. Comput. Harmon. Anal. 44(2), 476–496 (2018)

    Article  MathSciNet  Google Scholar 

  10. Fox, D.J.F.: The commutative nonassociative algebra of metric curvature tensors. arXiv:1901.04012

  11. Fox, D.J.F.: Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations. arXiv:0909.1897

  12. Fox, D.J.F.: Einstein-like geometric structures on surfaces. Ann. Sc. Norm. Super. Pisa Cl Sci. (5) XII(5), 499–585 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Fox, D.J.F.: Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein-Weyl and affine sphere equations, Extended abstracts Fall 2013—Geometrical Analysis, Type Theory, Homotopy Theory and Univalent Foundations, Trends Math. Res. Perspect. CRM Barc., vol. 3, Birkhäuser/Springer, pp. 15–19 (2015)

  14. Freudenthal, H.: Oktaven, Ausnahmegruppen und Oktavengeometrie. Geom. Dedicata 19(1), 7–63 (1985)

    Article  MathSciNet  Google Scholar 

  15. Harada, K.: On a commutative nonassociative algebra associated with a multiply transitive group. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), (1981), 843–849 (1982)

  16. Hsiang, W.-Y.: Remarks on closed minimal submanifolds in the standard Riemannian \(m\)-sphere. J. Differ. Geom. 1, 257–267 (1967)

    Article  MathSciNet  Google Scholar 

  17. Losik, M., Michor, P.W., Popov, V.L.: On polarizations in invariant theory. J. Algebra 301(1), 406–424 (2006)

    Article  MathSciNet  Google Scholar 

  18. Matsuo, A.: Norton’s trace formulae for the Griess algebra of a vertex operator algebra with larger symmetry. Commun. Math. Phys. 224(3), 565–591 (2001)

    Article  MathSciNet  Google Scholar 

  19. Nadirashvili, N., Tkachev, V., Vlăduţ, S.: Nonlinear Elliptic Equations and Nonassociative Algebras. Mathematical Surveys and Monographs, vol. 200. American Mathematical Society, Providence (2014)

    MATH  Google Scholar 

  20. Popov, V.L.: An analogue of M. Artin’s conjecture on invariants for nonassociative algebras, Lie groups and Lie algebras: E. B. Dynkin’s Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 169, Amer. Math. Soc., Providence, RI, pp. 121–143 (1995)

  21. Sato, M., Kimura, T.: A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J. 65, 1–155 (1977)

    Article  MathSciNet  Google Scholar 

  22. Smith, R.T.: Harmonic mappings of spheres. Am. J. Math. 97, 364–385 (1975)

    Article  MathSciNet  Google Scholar 

  23. Stein, E.M., Weiss, G.: Generalization of the Cauchy-Riemann equations and representations of the rotation group. Am. J. Math. 90, 163–196 (1968)

    Article  MathSciNet  Google Scholar 

  24. Tkachev, V.G.: The universality of one half in commutative nonassociative algebras with identities. arXiv:1808.03808

  25. Tkachev, V.G.: A generalization of Cartan’s theorem on isoparametric cubics. Proc. Am. Math. Soc. 138(8), 2889–2895 (2010)

    Article  MathSciNet  Google Scholar 

  26. Tkachev, V.G.: A Jordan algebra approach to the cubic eiconal equation. J. Algebra 419, 34–51 (2014)

    Article  MathSciNet  Google Scholar 

  27. Tkachev, V.G.: A correction of the decomposability result in a paper by Meyer-Neutsch. J. Algebra 504, 432–439 (2018)

    Article  MathSciNet  Google Scholar 

  28. Tkachev, V.G.: Hsiang algerbas and cubic minimal cones (in preparation) (2018)

  29. Tkachev, V.G., Miklyukov, V.M.: from dimension 8 to nonassociative algebras. Math. Phys. Comput. Simul. 22(2), 33–50 (2019)

    Article  MathSciNet  Google Scholar 

  30. Toth, G.: Eigenmaps and the space of minimal immersions between spheres. Indiana Univ. Math. J. 46(2), 637–658 (1997)

    Article  MathSciNet  Google Scholar 

  31. Vale, R., Waldron, S.: Tight frames and their symmetries. Constr. Approx. 21(1), 83–112 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Vale, R., Waldron, S.: The construction of \(G\)-invariant finite tight frames. J. Fourier Anal. Appl. 22(5), 1097–1120 (2016)

    Article  MathSciNet  Google Scholar 

  33. Wood, R.: Polynomial maps from spheres to spheres. Invent. Math. 5, 163–168 (1968)

    Article  MathSciNet  Google Scholar 

  34. Ye, K.: The stabilizer of immanants. Linear Algebra Appl. 435(5), 1085–1098 (2011)

    Article  MathSciNet  Google Scholar 

Download references

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  1. Departamento de Matemática Aplicada a la Ingeniería Industrial Escuela Técnica Superior de Ingeniería y Diseño Industrial, Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012, Madrid, Spain

    Daniel J. F. Fox

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Fox, D.J.F. Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic form. Anal.Math.Phys. 11, 43 (2021). https://doi.org/10.1007/s13324-020-00462-4

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