Analysis of the horizontal Laplacian for the Hopf fibration

We study the horizontal Laplacian Δ H associated to the Hopf fibration S3 → S2 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.