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We discuss certain geometric properties for area stationary currents and currents with integrable mean curvature, so called “enclosure theorems”. As a consequence, we obtain non-existence results for currents with connected support. Finally, we extend these results to currents in submanifolds and state a non-existence result for stationary currents in spheres.

Enclosure and non-existence theorems for area stationary currents and currents with mean curvature vector