Consider a two-dimensional Riemannian manifold homeomorphic to the sphere, with a defined metric.
Since we do not suppose that manifold to have a positive curvature, we are not in the hypotheses of the Pogorelov's uniqueness theorem. As a consequnce, in general, there are many possible isometric embeddings in $\mathbb R^3$ for that manifold.
The question is the following: is the maximum volume embedding of that manifold unique, up to rigid transformations?
The problem can be expressed in a more intuitive way: Suppose we have an inflatable object whose walls can be bent but not stretched. It can be seen as a closed manifold of known metric. If I inflate such an object until I reach the maximum volume, is the resulting object rigid? (suppose that there aren't narrow bottlenecks that could divide the object into two or more parts).