An isoperimetric inequality for the second non-zero eigenvalue of the   Laplacian on the projective plane
  We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane. For a metric of the unit area this eigenvalue is not greater than 20\pi. This value is attained in the limit by a sequence of metrics of area one on the projective plane. The limiting metric is singular and could be realized as a union of the projective plane and the sphere touching at a point, with standard metrics and the ratio of the areas 3:2. It is also proven that the multiplicity of the second non-zero eigenvalue on the projective plane is at most 6. 
Nikolai S. Nadirashvili, Alexei V. Penskoi
math.DG math.SP