Suppose that we have a convex cap, i.e., a convex surface in $R^3$ homeomorphic to a disk whose boundary lies in a plane. Reflect the cap through the plane of its boundary and glue it back to the original cap along the common boundary curve. If the closed surface we obtain is convex, then it is rigid by a theorem of Pogorelov, i.e., any surface which is isometric to it is congruent to it.

My question is: do we still have rigidity when the doubled surface is not convex? I do not know the answer even in the case where each cap is spherical. Here we want to look at isometric surfaces which are smooth except along the curve where the two pieces meet, so that we do not get trivial examples by reflecting a portion of the surface.

In general very little is known about rigidity of non-convex surfaces, but this situation is so simple that perhaps someone might have an idea about it. It seems to me that this is the simplest non-convex case.