Volume of a vortex and the Bradlow bound

We demonstrate that the geometric volume of a soliton coincides with the thermodynamical volume also for field theories with higher-dimensional vacuum manifolds (e.g., for gauged scalar field theories supporting vortices or monopoles), generalizing the recent results of Ref. [C. Adam, M. Haberichter, and A. Wereszczynski, Phys. Lett. B 754, 18 (2016).]. We apply this observation to understand Bradlow-type bounds for general Abelian gauge theories supporting vortices, as well as some thermodynamical aspects of said theories. In the case of SDiff Bogomolny-Prasad-Sommerfield (BPS) models (being examples of perfect fluid models) we show that the base-space independent geometric volume (area) of the vortex is exactly equal to the Bradlow volume (a minimal volume for which BPS soliton solutions exist). This volume can be finite for compactons or infinite for infinitely extended solitons (in flat Minkowski space-time).