I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
A comparison of current versus varifold representations of images that discusses their pros and cons: Current- and Varifold-Based Registration of Lung Vessel and Airway Trees
Varifolds are weaker objects than currents due to the lack of orientation of the tangent vector of the momenta used to represent a shape. For some objects where the orientation matters, such as velocity fields, only current representations are acceptable. For other objects, such as the airway trees studied in this paper, the orientation is not essential and varifolds increase the robustness when matching line segments with uncertain tangent orientation.
When matching a target to a template an error in the orientation of the tangent fails to produce a good match for the current representation (left panel), while the undetermined orientation of the varifold representation does produce a good match (right panel).
Image matching results using current and varifold representations. The circles show the branches that had opposite tangent orientations between the template and the target. The current-based registration fails (left panel) while the varifold-based registration succeeds (right panel).