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We consider a sequence of bubble converging minimal hypersurfaces, or H-CMChypersurfaces, in compact Riemannian manifolds without boundary, of dimension4, 5, 6 or 7, and prove upper semicontinuity of index plus nullity, for such abubble converging sequence. This complements the previously known lowersemicontinuity of index obtained by Buzano--Sharp, and Bourni--Sharp--Tinaglia.  The strategy of our proof is to analyse a weighted eigenvalue problem alongour sequence of hypersurfaces. This strategy is inspired by the recent work ofDa Lio--Gianocca--Rivi\`{e}re. A key aspect of our proof is making use of aLorentz--Sobolev inequality to study the behaviour of these weightedeigenfunctions on the neck regions along the sequence, as well as the index andnullity of our non-compact bubbles.

Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces