Spaces of harmonic maps of the projective plane to the four-dimensional sphere
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Journal of Geometry
Spaces of harmonic maps of the projective plane to the four-dimensional sphere Ravil Gabdurakhmanov Abstract. The spaces of harmonic maps of the projective plane to the fourdimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree less than 6 it was shown that such spaces are path-connected and an explicit parameterization of the canonical representatives was found. In addition, the last section provides comparisons with the known results for harmonic maps of the two-dimensional sphere to the four-dimensional sphere of harmonic degree less than 6. Mathematics Subject Classification. 58E20, 53C28, 53C43. Keywords. Harmonic map, twistor lift, projective plane.
1. Introduction Let φ : M → N be a map between Riemannian manifolds. We define its energy by the formula 1 E(φ) = |dφ(x)|2 dx, 2 M
where dφ(x) is the differential of φ at the point x ∈ M ; and dx is the volume element of M . Euler-Lagrange operator τ (φ) = div(dφ) associated with the functional E is called a tension field of φ. The map φ : M → N is said to be harmonic if its tension field vanishes identically i.e. φ is a critical point of E. Some particular cases of harmonic maps are well-known, i.e. • If dim M = 1, then the harmonic maps are the geodesics of N . • If N = R, they are harmonic functions on M . Supported in part by the Simons Foundation. 0123456789().: V,-vol
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R. Gabdurakhmanov
J. Geom.
• If dim M = 2, they include parametric representations of the minimal surfaces of N ; the energy is the Dirichlet-Douglas integral. The author’s interest to harmonic maps is motivated by the relationship between harmonic maps and isoperimetric inequalities for the Laplace operator eigenvalues. It was discovered by Nadirashvili [19] and El Soufi and Ilias [11] that these inequalities are closely connected to minimal and harmonic maps from surfaces to spheres S n . This connection permitted to completely solve recently the problem of isoperimetric inequalities for Laplace eigenvalues on the sphere S n , see the paper [15] for the general case and the previous papers [13,21,22] for particular cases, and on the projective plane, see the paper [14] for the general case and the previous papers [17] and [20] for particular cases. More information on this subject could be found in surveys [23] and [24]. There are many papers on the theory of the harmonic maps. This paper is based on the results of the famous works of E. Calabi [6,7], J. Barbosa [1], R. L. Bryant [5], and essentially uses ideas developed in works of J. Bolton and L. M. Woodward [2–4]. We also refer to important results on the topology of spaces of harmonic maps from S 2 to S 4 obtained by B. Loo in [18], M. Kotani in [16], and M. Furuta, M. A. Guest, M. Kotani, and Y. Ohnita in [12]. We use the following proposition and fundamental theorems. Proposition 1 [10]. An isometric immersion φ : (M, g) → (N, h) is minimal if and only if it is harmonic. Theo
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