Rigidity beyond the structural assumptions (P1)–(P3)

Determine whether rigidity holds for complete, minimal, stable, non-characteristic hypersurfaces in the second Heisenberg group without assuming conditions (P1), (P2), and (P3), and characterize how these structural conditions are related to minimality.

Background

A previous result cited in the paper classifies complete, minimal, stable, non-characteristic hypersurfaces in the second Heisenberg group under three additional assumptions concerning the horizontal second fundamental form and the fundamental function: (P1), (P2), and (P3). The result also requires a suitable area-growth condition.

Before presenting the counterexamples constructed in the paper, the authors explicitly state that it remained unresolved whether the rigidity conclusion persists after removing these assumptions and how the assumptions themselves relate to minimality. The paper subsequently gives counterexamples that answer these questions negatively, but the passage identifies the problem as an open problem left by the earlier work.