  A supersymmetric solution of 5d supergravity may admit an `evanescent
ergosurface': a timelike hypersurface such that the canonical Killing vector
field is timelike everywhere except on this hypersurface. The hyperk\"ahler
`base space' of such a solution is `ambipolar', changing signature from
$(++++)$ to $(----)$ across a hypersurface. In this paper, we determine how the
hyperk\"ahler structure must degenerate at the hypersurface in order for the 5d
solution to remain smooth. This leads us to a definition of an ambipolar
hyperk\"ahler manifold which generalizes the recently-defined notion of a
`folded' hyperk\"ahler manifold. We prove that such manifolds can be
constructed from `initial' data prescribed on the hypersurface. We present an
`initial value' construction of supersymmetric solutions of 5d supergravity, in
which such solutions are determined by data prescribed on a timelike
hypersurface, both for the generic case and for the case of an evanescent
ergosurface.
