Concentration at umbilic boundary points

Determine whether blow-up or concentration can occur at boundary points where the trace-free second fundamental form vanishes, beyond the established compactness results for scalar-flat boundary Yamabe metrics when that tensor is nowhere zero.

Background

The paper reviews Almaraz’s compactness theorem for scalar-flat metrics with boundary under the assumption that the trace-free second fundamental form is nowhere zero. It then notes that the known results do not resolve concentration at points where this tensor vanishes. The present paper constructs noncompactness examples with umbilic boundary in sufficiently high dimensions, but the quoted passage identifies the general possibility of concentration at such points as unresolved in the surrounding literature.