Rigidity and Classification of Manifolds with Maximal Boundary Static Kernels

Abstract

Boundary static potentials arise from the adjoint equations associated with simultaneous variations of scalar curvature and boundary mean curvature. On a connected manifold of dimension $n\ge 2$ with nonempty smooth boundary, the space of boundary static potentials has dimension at most $n$. We classify all complete manifolds attaining this bound. For $n\ge 3$, every such manifold is isometric to a domain in a simply connected space form and has at most two boundary components. In dimension two, the classification also includes coverings of annuli and exteriors, as well as spherical Möbius bands. Finally, for $n\ge 3$, the dimension hypothesis is sharp for constant-curvature rigidity, and for $n\ge 2$, it is sharp for the bound on the number of boundary components.