  We discuss the duality, conjectured in earlier work, between the wave
function of the multiverse and a 3D Euclidean theory on the future boundary of
spacetime. In particular, we discuss the choice of the boundary metric and the
relation between the UV cutoff scale xi on the boundary and the hypersurfaces
Sigma on which the wave function is defined in the bulk. We propose that in the
limit of xi going to 0 these hypersurfaces should be used as cutoff surfaces in
the multiverse measure. Furthermore, we argue that in the inflating regions of
spacetime with a slowly varying Hubble rate H the hypersurfaces Sigma are
surfaces of constant comoving apparent horizon (CAH). Finally, we introduce a
measure prescription (called CAH+) which appears to have no pathological
features and coincides with the constant CAH cutoff in regions of slowly
varying H.
