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Abstract
This paper is concerned with the low eigenvalues of closed surfaces in , of given measure, which are topologically equivalent to a sphere. Our aim is to obtain an isoprimetric inequality giving an upper bound for the product of the first three non-trivial eigenvalues of a convex closed surface topologically equivalent to a sphere. Moreover, we will also derive some lower bounds for the first non-trivial eigenvalue of the regular octahedron and icosahedron.
Acknowledgements
The authors would like to thank the anonymous referees for their valuable suggestions to improve clarity in several points.
Notes
No potential conflict of interest was reported by the authors.
Dedicated to Prof. Shigeru Sakaguchi on the occasion of his 60th birthday.
Additional information
Funding
This work was supported by a NSERC (Canada) research grant (EG-1508: Principes du maximum, inégalités isopérimétriques et problèmes mal posés).