Harmonic Morphisms from Fefferman Spaces

We study a ramification of a phenomenon discovered by Baird and Eells (in: Looijenga et al (eds) Geometry Symposium Utrecht 1980. Lecture Notes in Mathematics, Springer, Berlin, 1981) i.e. that non-constant harmonic morphisms \(\Phi : {{\mathfrak {M}}}^{\textrm{N}} \rightarrow N^2\) from a \(\mathrm N\)-dimensional (\(\textrm{N} \ge 3\)) Riemannian manifold \({{\mathfrak {M}}}^{\textrm{N}}\), into a Riemann surface \(N^2\), can be characterized as those horizontally weakly conformal maps having minimal fibres. We recover Baird–Eells’ result for \(S^1\) invariant harmonic morphisms \(\Phi : {{\mathfrak {M}}}^{2n+2} \rightarrow N^2\) from a class of Lorentzian manifolds arising as total spaces \({{\mathfrak {M}}} = C(M)\) of canonical circle bundles \(S^1 \rightarrow {{\mathfrak {M}}} \rightarrow M\) over strictly pseudoconvex CR manifolds \(M^{2n+1}\). The corresponding base maps \(\phi : M^{2n+1} \rightarrow N^2\) are shown to satisfy \(\lim _{\epsilon \rightarrow 0^+} \, \pi _{{{\mathscr {H}}}^\phi } \, \mu ^{{{\mathscr {V}}}^\phi }_\epsilon = 0\), where \(\mu ^{{{\mathscr {V}}}^\phi }_\epsilon \) is the mean curvature vector of the vertical distribution \({{\mathscr {V}}}^\phi = \textrm{Ker} (d \phi )\) on the Riemannian manifold \((M, \, g_\epsilon )\), and \(\{ g_\epsilon \}_{0< \epsilon < 1}\) is a family of contractions of the Levi form of the pseudohermitian manifold \((M, \, \theta )\).