Upper Bounds for the First Eigenvalue of the Laplacian on Non-Orientable Surfaces
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Abstract
In 1980, Yang and Yau [20] proved the celebrated upper bound for the first eigenvalue on an orientable surface of genus |$\gamma $|. Later Li and Yau [14] gave a simple proof of this bound by introducing the concept of conformal volume of a Riemannian manifold. In the same paper, they proposed an approach for obtaining a similar estimate for non-orientable surfaces. In the present paper, we formalize their argument and improve the bounds stated in [14].
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
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| Bibliometrics | 0.000 | 0.001 |
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| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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