  Moduli spaces - finite-dimensional, collective coordinate manifolds - for
kinks and antikinks in $\phi^4$ theory and sine-Gordon theory are reconsidered.
The field theory Lagrangian restricted to moduli space defines a reduced
Lagrangian, combining a potential with a kinetic term that can be interpreted
as a Riemannian metric on moduli space. Moduli spaces should be metrically
complete, or have an infinite potential on their boundary. Examples are
constructed for both kink-antikink and kink-antikink-kink configurations. The
naive position coordinates of the kinks and antikinks sometimes need to be
extended from real to imaginary values, although the field remains real. The
previously discussed null-vector problem for the shape modes of $\phi^4$ kinks
is resolved by a better coordinate choice. In sine-Gordon theory, moduli spaces
can be constructed using exact solutions at the critical energy separating
scattering and breather (or wobble) solutions; here, energy conservation
relates the metric and potential. The reduced dynamics on these moduli spaces
accurately reproduces properties of the exact solutions over a range of
energies.
