ИСТИНА

The question of description of metrics extremal for eigenvalues of Laplace operator is one of the classical questions of spectral geometry. Recently it turned out that the extremal metrics are exactly the metrics on minimal submanifolds in spheres. The condition of minimality of a submanifold is a complicated PDE that can be solved only in some special cases. It turned out that the symmetry reduction theory for minimal submanifolds developed by Hsiang and Lawson could give us new examples of minimal S^1-invariant tori and Klein bottles in spheres such that we can investigate extremal spectral properties of their metrics.