About a Proposition on Escobar's paper "The Geometry of the First Non-zero Stekloff Eigenvalue"
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909134920613888 |
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| author | Quiñones, Leoncio Rodriguez |
| author_facet | Quiñones, Leoncio Rodriguez |
| contents | Let $(M^{2},g_{0})$ be a compact manifold with boundary, and let $g$ and $g_{0}$ be conformally related by $g=e^{2f}g_{0}$. We show that the inequality $$ν_{1}(g)\geq\Big(\max_{x\in\partial M}e^{-f(x)}\Big)ν_{1}(g_{0})$$ stated in Proposition 2 in [1], is only possible when the equality is achieved. In order to achieve such equality, it is required that the function $f$ be constant on $\partial M$, as it is mentioned in Remark 3 also in [1]. Hence, the scope of this inequality is less broad than the one suggested by the Proposition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_16930 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | About a Proposition on Escobar's paper "The Geometry of the First Non-zero Stekloff Eigenvalue" Quiñones, Leoncio Rodriguez Differential Geometry Spectral Theory 58J05, 58J32, 58J50 Let $(M^{2},g_{0})$ be a compact manifold with boundary, and let $g$ and $g_{0}$ be conformally related by $g=e^{2f}g_{0}$. We show that the inequality $$ν_{1}(g)\geq\Big(\max_{x\in\partial M}e^{-f(x)}\Big)ν_{1}(g_{0})$$ stated in Proposition 2 in [1], is only possible when the equality is achieved. In order to achieve such equality, it is required that the function $f$ be constant on $\partial M$, as it is mentioned in Remark 3 also in [1]. Hence, the scope of this inequality is less broad than the one suggested by the Proposition. |
| title | About a Proposition on Escobar's paper "The Geometry of the First Non-zero Stekloff Eigenvalue" |
| topic | Differential Geometry Spectral Theory 58J05, 58J32, 58J50 |
| url | https://arxiv.org/abs/2311.16930 |