Analysis of the horizontal Laplacian for the Hopf fibration

We study the horizontal Laplacian $\Delta^H$ 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 $\Delta^H$. We express the Green functions for associated Poisson semigroups and obtain bounds for their contraction properties and Sobolev inequalities for $\Delta^H$. The bounds and inequalities improve as $|k|$ increases.