Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces

We consider a sequence of bubble converging minimal hypersurfaces, or H-CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension 4, 5, 6 or 7, and prove upper semicontinuity of index plus nullity, for such a bubble converging sequence. This complements the previously known lower semicontinuity of index obtained by Buzano–Sharp, and Bourni–Sharp–Tinaglia. The strategy of our proof is to analyse a weighted eigenvalue problem along our sequence of hypersurfaces. This strategy is inspired by the recent work of Da Lio–Gianocca–Rivière. A key aspect of our proof is making use of a Lorentz–Sobolev inequality to study the behaviour of these weighted eigenfunctions on the neck regions along the sequence, as well as the index and nullity of our non-compact bubbles.