An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane
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.
Keywords
Cite
@article{arxiv.1608.07334,
title = {An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective plane},
author = {Nikolai S. Nadirashvili and Alexei V. Penskoi},
journal= {arXiv preprint arXiv:1608.07334},
year = {2018}
}
Comments
28 pages, v2: minor errors corrected, v3: major revision, the proof is simplified, v4: minor errors corrected, some proofs and the expository part are extended, v5: the statement of the main theorem is made more precise