## Surfaces minimales dans les sphères. (Minimal surfaces in spheres).(French)Zbl 0635.53037

Théorie des variétés minimales et applications, Sémin. Palaiseau/France 1983/1984, Astérisque 154-155, 115-130 (1988).
Summary: [For the entire collection see Zbl 0635.53001.]
It is a classical and elementary fact that any holomorphic map $$\phi$$ from a Riemann surface $$\Sigma$$ in a Kähler manifold X is minimal and that, $$\pi$$ being a Riemannian submersion from X onto a manifold Y, $$\pi\circ \phi$$ is minimal as soon as $$\phi$$ is horizontal with respect to $$\pi$$. A converse due to E. Calabi is proven here in the case where $$\Sigma =S$$ 2, $$Y=S^{n-1}\subset {\mathbb{R}}^ n.$$ One constructs a Riemannian submersion from a submanifold X of $${\mathbb{C}}P$$ N onto $$S^{N- 1}$$ and a holomorphic horizontal lift of $$\phi$$. From this one can deduce that the image of a minimal immersion from S 2 into $$S^{n-1}$$ is either contained in an equator, or has a quantified area $$4\pi$$ k for $$k\geq \left( \begin{matrix} m+1\\ 2\end{matrix} \right)$$ (Calabi-Barbosa).

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Zbl 0635.53001