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Analysis of the horizontal Laplacian for the Hopf fibration

Robert O. Bauer
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Abstract

We study the horizontal Laplacian $Δ^H$ associated to the Hopf fibration $S^3\to S^2$ with arbitrary Chern number $k$. We use representation theory to calculate the spectrum, describe the heat kernel and obtain the complete heat trace asymptotics of $Δ^H$. We express the Green functions for associated Poisson semigroups and obtain bounds for their contraction properties and Sobolev inequalities for $Δ^H$. The bounds and inequalities improve as $|k|$ increases.

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Robert O. Bauer. 2002-09-11. Analysis of the horizontal Laplacian for the Hopf fibration. https://arxiv.org/abs/math/0209127

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