Spectre et géométrie conforme des variétés compactes à bord
2014 ◽
Vol 150
(12)
◽
pp. 2112-2126
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Keyword(s):
AbstractWe prove that on any compact manifold $M^{n}$ with boundary, there exists a conformal class $C$ such that for any Riemannian metric $g\in C$ of unit volume, the first positive eigenvalue of the Neumann Laplacian satisfies ${\it\lambda}_{1}(M^{n},g)<n\,\text{Vol}(S^{n},g_{\text{can}})^{2/n}$. We also prove a similar inequality for the first positive Steklov eigenvalue. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of $(M,C)$ is $\text{Vol}(S^{n},g_{\text{can}})$, and that the Friedlander–Nadirashvili invariant and the Möbius volume of $M$ are equal to those of the sphere. If $M$ is a domain in a space form, $C$ is the conformal class of the canonical metric.
2006 ◽
Vol 03
(05n06)
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pp. 833-844
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2019 ◽
Vol 72
(4)
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pp. 1024-1043
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2014 ◽
Vol 07
(01)
◽
pp. 23-46
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2019 ◽
Vol 21
(03)
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pp. 1850021
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2009 ◽
Vol 61
(3)
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pp. 548-565
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Keyword(s):