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Article

An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane

Geometric and Functional Analysis. 2018. Vol. 28. No. 5. P. 1368-1393.
Penskoi Alexei V., Nadirashvili N. S.

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 unit area this eigenvalue is not greater than 20π. 20π. 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.