  We propose a novel framework to solve PDEs on moving manifolds, where the
evolving surface is represented by a moving point cloud. This has the advantage
of avoiding the need to discretize the bulk volume around the surface, while
also avoiding the need to have a global mesh. Distortions in the point cloud as
a result of the movement are fixed by local adaptation. We first establish a
comprehensive Lagrangian framework for arbitrary movement of curves and
surfaces given by point clouds. Collision detection algorithms between point
cloud surfaces are introduced, which also allow the handling of evolving
manifolds with topological changes. We then couple this Lagrangian framework
with a meshfree Generalized Finite Difference Method (GFDM) to approximate
surface differential operators, which together give a method to solve PDEs on
evolving manifolds. The applicability of this method is illustrated with a
range of numerical examples, which include advection-diffusion equations with
large deformations of the surface, curvature dependent geometric motion, and
wave equations on evolving surfaces.
