On the moduli space of superminimal surfaces in spheres
It is known [\textit{B. Loo}, Trans. Am. Math. Soc. 313, No. 1, 81--102 (1989; Zbl 0686.58009)] that the dimension of the moduli space of harmonic maps \(f : S^2 \to S^4\) of degree \(d\) in \(S^4\) is \(2d + 4\). In this case the twistor space \({\mathcal L}_2 = \mathrm{SO}(5)/\mathrm{SO}(2)\) is isomorphic to the complex projective space \({\mathbb P}^3\) and so one can characterize twistor lifts of minimal maps as solutions in \({\mathbb P}^3\) of a single differential equation. The paper under review deals with the moduli space of superminimal surfaces into higher dimensional sphere \(S^{2n}\). A superminimal surface in \(S^{2n}\) can be defined as a harmonic map \(f: M_g \to S^{2n}\), where \(M_g\) is a Riemann surface of genus \(g\), whose osculating plane is totally isotropic (i.e., the symmetric complex product of any two vectors in that subspace is zero). In this case the twistor space \({\mathcal L}_n\) of \(S^{2n}\) becomes \(\mathrm{SO}(2n+1)/\mathrm{SO}(n)\) whose dimension is \(\displaystyle{N = \frac{n(n+1)}{2}}\). Although \({\mathcal L}_n\) and \({\mathbb P}^N\) are not isomorphic to each other, they are birationally equivalent. Using a birational correspondence between the twistor space of \(S^{2n}\) and projective space, the author described, up to birational equivalence, the moduli space of superminimal surfaces in \(S^{2n}\) of degree \(d\) as curves of degree \(d\) in projective space satisfying a certain differential system. Using this description, the author also showed that the moduli space of linearly full maps (i.e., not contained in a proper subsphere of \(S^{2n}\)) is nonempty for sufficiently large degree and the dimension of this moduli space for \(n = 3\) and genus \(0\) is greater than or equal to \(2d + 9\). The author also gave a direct, simple proof of the connectedness of the moduli space of superminimal surfaces in \(S^{2n}\) of degree \(d\).
scientific article; zbMATH DE number 2012578