  Transport and mixing in dynamical systems are important properties for many
physical, chemical, biological, and engineering processes. The detection of
transport barriers for dynamics with general time dependence is a difficult,
but important problem, because such barriers control how rapidly different
parts of phase space (which might correspond to different chemical or
biological agents) interact. The key factor is the growth of interfaces that
partition phase space into separate regions. The paper [Froyland, Nonlinearity
2015] introduced the notion of \textit{dynamic isoperimetry}: the study of sets
with persistently small boundary size (the interface) relative to enclosed
volume, when evolved by the dynamics. Sets with this minimal boundary size to
volume ratio were identified as level sets of dominant eigenfunctions of a
\textit{dynamic Laplace operator}.
  In this present work we extend the results of [Froyland, Nonlinearity 2015]
to the situation where the dynamics (i) is not necessarily volume-preserving,
(ii) acts on initial agent concentrations different from uniform
concentrations, and (iii) occurs on a possibly curved phase space. Our main
results include generalised versions of the dynamic isoperimetric problem, the
dynamic Laplacian, Cheeger's inequality, and the Federer-Fleming theorem. We
illustrate the computational approach with some simple numerical examples.
