A reference mesh is a (discrete) domain on which the deformation fields sampled by a Gaussian process is applied. The mean mesh is the reference mesh deformed by the mean deformation field.
A sampled mesh or shape = the reference mesh deformed by a sampled deformation field.
The mean mesh or shape =
the reference mesh deformed by the mean deformation field.
Note that the deformation field is a random filed (like a random vector) so it has a mean.
This is as per my knowledge. If contains error, please correct me.