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 $\Phi: \mathfrak{M}^N \to N^2$ from a N-dimensional ($N \geq 3$) Riemannian manifold $\mathfrak{M}^N$, 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 $\Phi : \mathfrak{M}^{2n+2} \to N^2$ from a class of Lorentzian manifolds arising as total spaces $\mathfrak{M}=C(M)$ of canonical circle bundles $S^1 \to \mathfrak{M} \to M$ over strictly pseudoconvex CR manifolds $M^{2n+1}$. The corresponding base maps $\phi : M^{2n+1} \to N^2$ are shown to satisfy $\lim_{\epsilon \to 0^+} \pi_{\mathcal{H}^\phi} \mu_\epsilon^{\mathcal{V}^\phi}= 0$, where $\mu_\epsilon^{\mathcal{V}^\phi}$ is the mean curvature vector of the vertical distribution $\mathcal{V}^\phi= {\rm Ker} (d\phi)$ 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 $(M, \theta)$.
Harmonic Morphisms from Fefferman Spaces
Sorin Dragomir
;Francesco Esposito;Eric Loubeau
2024-01-01
Abstract
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 $\Phi: \mathfrak{M}^N \to N^2$ from a N-dimensional ($N \geq 3$) Riemannian manifold $\mathfrak{M}^N$, 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 $\Phi : \mathfrak{M}^{2n+2} \to N^2$ from a class of Lorentzian manifolds arising as total spaces $\mathfrak{M}=C(M)$ of canonical circle bundles $S^1 \to \mathfrak{M} \to M$ over strictly pseudoconvex CR manifolds $M^{2n+1}$. The corresponding base maps $\phi : M^{2n+1} \to N^2$ are shown to satisfy $\lim_{\epsilon \to 0^+} \pi_{\mathcal{H}^\phi} \mu_\epsilon^{\mathcal{V}^\phi}= 0$, where $\mu_\epsilon^{\mathcal{V}^\phi}$ is the mean curvature vector of the vertical distribution $\mathcal{V}^\phi= {\rm Ker} (d\phi)$ 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 $(M, \theta)$.File | Dimensione | Formato | |
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