As is well-known, the real rank of a simple Lie group that acts conformally on a pseudo-Riemannian manifold is bounded by means of the signature of the manifold. We give a precise description of the action whenever the real rank of the group reaches that bound, assuming the action is minimal.

In this article we introduce and prove the solvability of the gauge-fixing constant scalar curvature equations on ruled Kaehler manifolds. We prove that when some lifting conditions for holomorphic vector fields on the base manifold are satisfied the solutions for the gauge-fixing constant scalar curvature equations are actually solutions for the constant scalar curvature equations provided the corresponding Futaki invariants vanish.

We prove that the error term $R(\lambda)$ in Weyl's law is $\mathcal O_{\epsilon}(\lambda^{5/6 + \epsilon})$ for certain three-dimensional Heisenberg manifolds. We also show that the $L^2$-norm of the Weyl error term integrated over the moduli space of left-invariant Heisenberg metrics is $\ll \lambda^{3/4 + \epsilon}$. We conjecture that $R(\lambda) = \mathcal{O}_{\epsilon}(\lambda^{3/4 + \epsilon})$ is a sharp deterministic upper bound for Heisenberg three-manifolds.

It is well-known that any 4-dimensional hyperkähler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on • ³. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain 'multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kähler quotients of • P^m—1 by an (m — 2)-torus studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kähler quotients of • P² (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.