ИСТИНА

An isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane is proven. For a metric of area 1 this eigenvalue is not greater than 20\pi. 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. It is also proven that the multiplicity of the second non-zero eigenvalue on the projective plane is at most 6. The talk is based on a joint work with N. S. Nadirashvili.