An isoperimetric inequality for the second non-zero eigenvalue of the Laplacian on the projective planeстатья
Статья опубликована в высокорейтинговом журнале
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Аннотация: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.