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Reilly's type inequality for the Laplaci...
Domingo-Juan, M. Car...

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Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF by Domingo-Juan, M. Carmen ( Author )
Lemonjar Open Access
N.A
04-03-2015
Let (M¯,<,>,eψ) be a Riemannian manifold with a density, and let M be a closed n-dimensional submanifold of M¯ with the induced metric and density. We give an upper bound on the first eigenvalue λ1 of the closed eigenvalue problem for Δψ (the Laplacian on M associated to the density) in terms of the average of the norm of the vector H⃗ ψ+∇¯ with respect to the volume form induced by the density, where H⃗ ψ is the mean curvature of M associated to the density eψ. When M¯=Rn+k or M¯=Sn+k−1, the equality between λ1 and its bound implies that eψ is a Gaussian density (ψ(x)=C2|x|2, C<0), and M is a shrinker for the mean curvature flow (MCF) on Rn+k. We prove also that λ1=−C on the standard shrinker torus of revolution. Based on this and on the Yau's conjecture on the first eigenvalue of minimal submanifolds of Sn, we conjecture that the equality λ1=−C is true for all the shrinkers of MCF in Rn+k.
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Article
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36.88 KB
English
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MYR 0.01
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http://arxiv.org/abs/1503.01332
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