  We present a novel particle management method using the Characteristic
Mapping framework. In the context of explicit evolution of parametrized curves
and surfaces, the surface distribution of marker points created from sampling
the parametric space is controlled by the area element of the parametrization
function. As the surface evolves, the area element becomes uneven and the
sampling, suboptimal. In this method we maintain the quality of the sampling by
pre-composition of the parametrization with a deformation map of the parametric
space. This deformation is generated by the velocity field associated to the
diffusion process on the space of probability distributions and induces a
uniform redistribution of the marker points. We also exploit the semigroup
property of the heat equation to generate a submap decomposition of the
deformation map which provides an efficient way of maintaining evenly
distributed marker points on curves and surfaces undergoing extensive
deformations.
