= Solution
An <ergoregion of a stationary spacetime> is where the stationary <Killing vector field>, normalized as time translation at infinity, is spacelike; a fixed-coordinate stationary observer cannot remain timelike there. For the <Kerr metric>, $k^2=-(1-2Mr/\Sigma)$, so the outer stationary-limit surface is
$$
r_{\rm E}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.
$$
For a subextremal rotating hole $r_{\rm E}\geq r_+=M+\sqrt{M^2-a^2}$, with equality only at the poles. At the equator $r_{\rm E}=2M>r_+$. \b[The ergosphere touches the horizon on the rotation axis and is outside it elsewhere.] Its boundary is not the horizon, whose generator is $k+\Omega_Hm$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-58-kerr-surfaces.png]
{title=Kerr horizon and outer stationary-limit surface in an oblate-coordinate meridional sketch}
The drawing uses oblate Cartesian coordinates to display the intersection and is not an isometric embedding of the spatial geometry.
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