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).

anyclosed smooth surface in $\mathbb R^3$ is rigid. There was a similar conjecture for polyhedra, but Connelly found a counterexample. $\endgroup$ – Deane Yang Jul 15 '19 at 13:38starshaped). Anyway, the point is, after inflating an object with non-positive curvature, some parts of it may touch each other, no? $\endgroup$ – Pietro Majer Jul 15 '19 at 16:029more comments