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

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.

### 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