Minimal submanifolds of low cohomogeneity.

*(English)*Zbl 0219.53045Let \(N\) be an invariant \(n\)-dimensional Riemannian subspace of the Riemannian \(G\)-space \(M\) where \(G\) is a compact Lie subgroup of the full isometry group of \(M\) leaving \(N\) invariant such that there is a principal orbit of the \(G\)-action on \(M\) which is contained in \(N\). Let \(\nu\) denote the dimension of this principal orbit. The number \(n-\nu\) is called the cohomogeneity of \(N\) in the \(G\)-space \(M\). Continuing previous works of the first author [Proc. Natl. Acad. Sci. USA 56, 5–6 (1966; Zbl 0178.559) and J. Differ. Geom. 1, 257–267 (1967; Zbl 0168.429)] the authors study minimal submanifolds of homogeneous spaces and give some classification of low cohomogeneity minimal submanifolds of spheres in the case of compact subgroups of isometries of \(\mathbb R^n\). The results of this paper are too many and too complicated to be given here in full detail.

In the first chapter the authors regard actions of a compact connected group \(G\) of isometries of a Riemannian manifold \(M\). They show that a \(G\)-invariant submanifold \(N\) of \(M\) is minimal if and only if the volume of \(N\) is stationary with respect to all compactly supported \(G\)-equivariant variations of \(N\). Using well-known theorems from the theory of differentiable transformation groups they get a natural stratification of the singular set \(M_s\) of the \(G\)-action (= all points of \(M\) which are not contained in a principal \(G\)-orbit) by minimal submanifolds. Providing the orbit space \((M - M_s)/G\) with an appropriate metric \(g_k\) they reduce the study of minimal \(G\)-invariant submanifolds of \(M\) of cohomogeneity \(k\) to the study of minimal submanifolds of \((M - M_s)/G\). As a consequence they get a proof of the following theorem stated in the first reference of the first author (without proof): Every compact homogeneous space can be minimally immersed into \(S^n\). There are given further applications producing many new examples of homogeneous minimal submanifolds.

The second chapter serves as a preparation of the study of minimal submanifolds of spheres of low cohomogeneity. The authors classify the actions of compact linear groups of cohomogeneity 2 or 3 on \(\mathbb R^n\) together with their orbit structures and the appropriate metrIcs.

The topic of the third chapter are closed cohomogeneity-one minimal hypersurfaces of \(S^n\). By the results of the first chapter the problem of understanding the behaviour and classifying such hypersurfaces has been reduced to the study of closed geodesics of the corresponding orbit space where “closed” means compact such that the boundary points belong to the regular part of \(S^n\).

There are investigated two cases:

1) The orbit space is a disk with a rotationally invariant metric.

2) The orbit space is a region of the Euclidean 2-sphere bounded by two or three great circular arcs.

A complete classification of closed cohomogeneity-one minimal hypersurfaces in \(S^{n+1}\) (resp. \(S^{2n-1}\)) with respect to the action \(\rho_n\oplus 2\vartheta\) (resp. \(\rho_n\oplus \rho_n\)) of \(\mathrm{SO}(n)\) (where \(\rho_n\) denotes the standard and \(\vartheta\) the trivial representation of \(\mathrm{SO}(n)\)) is given.

In the final chapter the authors use the above methods to present a complete classification of low cohomogeneity minimal surfaces in \(S^3\). For related material and further applications see the papers of the second author in [Ann. Math (2) 92, 335–374 (1970; Zbl 0205.52001) and J. Differ. Geom. 4, 349–357 (1970; Zbl 0199.56401)] and a paper of the second author, “The equivariant Plateau problem and interior regularity” [Trans. Am. Math. Soc. 173, 231–249 (1972; Zbl 0279.49043)].

In the first chapter the authors regard actions of a compact connected group \(G\) of isometries of a Riemannian manifold \(M\). They show that a \(G\)-invariant submanifold \(N\) of \(M\) is minimal if and only if the volume of \(N\) is stationary with respect to all compactly supported \(G\)-equivariant variations of \(N\). Using well-known theorems from the theory of differentiable transformation groups they get a natural stratification of the singular set \(M_s\) of the \(G\)-action (= all points of \(M\) which are not contained in a principal \(G\)-orbit) by minimal submanifolds. Providing the orbit space \((M - M_s)/G\) with an appropriate metric \(g_k\) they reduce the study of minimal \(G\)-invariant submanifolds of \(M\) of cohomogeneity \(k\) to the study of minimal submanifolds of \((M - M_s)/G\). As a consequence they get a proof of the following theorem stated in the first reference of the first author (without proof): Every compact homogeneous space can be minimally immersed into \(S^n\). There are given further applications producing many new examples of homogeneous minimal submanifolds.

The second chapter serves as a preparation of the study of minimal submanifolds of spheres of low cohomogeneity. The authors classify the actions of compact linear groups of cohomogeneity 2 or 3 on \(\mathbb R^n\) together with their orbit structures and the appropriate metrIcs.

The topic of the third chapter are closed cohomogeneity-one minimal hypersurfaces of \(S^n\). By the results of the first chapter the problem of understanding the behaviour and classifying such hypersurfaces has been reduced to the study of closed geodesics of the corresponding orbit space where “closed” means compact such that the boundary points belong to the regular part of \(S^n\).

There are investigated two cases:

1) The orbit space is a disk with a rotationally invariant metric.

2) The orbit space is a region of the Euclidean 2-sphere bounded by two or three great circular arcs.

A complete classification of closed cohomogeneity-one minimal hypersurfaces in \(S^{n+1}\) (resp. \(S^{2n-1}\)) with respect to the action \(\rho_n\oplus 2\vartheta\) (resp. \(\rho_n\oplus \rho_n\)) of \(\mathrm{SO}(n)\) (where \(\rho_n\) denotes the standard and \(\vartheta\) the trivial representation of \(\mathrm{SO}(n)\)) is given.

In the final chapter the authors use the above methods to present a complete classification of low cohomogeneity minimal surfaces in \(S^3\). For related material and further applications see the papers of the second author in [Ann. Math (2) 92, 335–374 (1970; Zbl 0205.52001) and J. Differ. Geom. 4, 349–357 (1970; Zbl 0199.56401)] and a paper of the second author, “The equivariant Plateau problem and interior regularity” [Trans. Am. Math. Soc. 173, 231–249 (1972; Zbl 0279.49043)].

Reviewer: Bernd Wegner (Berlin)