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).


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


Zbl 0635.53001