On surfaces of low genus whose twistor lifts are harmonic sections. (English) Zbl 1198.53048

Hasegawa, Kazuyuki

J. Geom. Symmetry Phys. 17, 35-43 (2010).

The author considers the twistor space \(Z\) of an oriented four-dimensional Riemannian manifold \(\tilde{M}\) with the canonical (“product”) Riemannian metric, and the canonical almost complex structure. Each immersed surface \(M\) on \(\tilde{M}\) has a natural twistor lift to \(Z\), and it is called a superminimal surface (twistor holomorphic surface, respectively) if its twistor lift is a horizontal map (holomorphic map, respectively) from \(M\) to \(Z\). The twistor lift is a harmonic section of \(Z\) if it is a stationary map for the energy functional among all sections of the twistor space. Supposing \(\tilde{M}\) is a hyper-Kähler manifold, and \(M\) is a compact surface of genus zero, if the twistor lift is a harmonic section of \(Z\), the author concludes that \(M\) is respectively a non-superminimal minimal surface, a superminimal surface, and a non-superminimal twistor holomorphic surface when \(\chi(T^{\bot}M)\geq 4\), \(\chi(T^{\bot}M)=2\), and \(\chi(T^{\bot}M)\leq 0\), respectively. This completely determines all genus zero surfaces with harmonic twistor lifts, since \(\chi(T^{\bot}M)\) is an even integer for \(\tilde{M}\) a hyper-Kähler manifold. This is applied to the case \(\tilde{M}=\mathbb{R}^4\) obtaining some conclusions by using the fact that no compact minimal submanifold exists, or to obtain an extension of Hopf’s theorem for a constant mean curvature surface of genus zero to surfaces with parallel mean curvature.
As another application in the case \(\tilde{M}=\mathbb{C}^2\), the author obtains the result of Castro and Urbano that, if \(M\) is a compact Lagrangian surface of genus zero and the Maslov form on \(M\) is conformal, then \(M\) is congruent to the Whitney immersion.

Reviewer: Isabel Salavessa (Lisboa)

MSC:

53C26	Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C28	Twistor methods in differential geometry
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40	Global submanifolds
58E20	Harmonic maps, etc.

Keywords:

harmonic sections; superminimal; twistor lift; hyper-Kähler