We discuss certain quantitative geometric properties of energy stationary currents describing minimal surfaces under gravitational forces. Enclosure theorems give statements about the confinement of the support of currents to certain enclosing sets on the basis that one knows something about the position of their boundaries. These results are closely related to non-existence theorems for currents with connected support. Finally, we define a weak formulation in the theory of varifolds for the curvature flow associated to this energy functional. We extend the enclosure results to the flow and discuss several comparison principles.
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