ИСТИНА

The first subject of this talk is an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane (based on a joint paper with N. Nadirashvili). For a metric of area 1 this eigenvalue is not greater than 20π. This value could be attained as a limit on a sequence of metrics of area 1 on the projective plane converging to a singular metric on the projective plane and the sphere with standard metrics touching in a point such that the ratio of the areas of the projective plane and the sphere is 3:2. The second subject of this talk is a very recent result (joint paper with M. Karpukhin, N. Nadirashvili and I. Polterovich) about an isoperimetric inequality for Laplace eigenvalues on S^2. For a metric of area 1 the k-th eigenvalue is not greater than 8πk. This value could be attained as a limit on a sequence of metrics of area 1 on the sphere converging to a singular metric on k spheres with standard metrics of equal radius touching in a point.