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.