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

TL;DR

This paper establishes an isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the real projective plane, identifying the maximal value and characterizing the extremal metric.

Contribution

It proves an upper bound of 20π for the second eigenvalue on the projective plane and describes the limiting singular metric achieving this bound.

Findings

The second eigenvalue is bounded above by 20π for unit area metrics.

The extremal metric is a union of the projective plane and sphere touching at a point.

The multiplicity of the second eigenvalue is at most 6.

Abstract

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.