Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces

Abstract We consider a sequence, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>M</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:mi>ℕ</m:mi> </m:mrow> </m:msub> </m:math> {\{M_{k}\}_{k\in\mathbb{N}}} , of bubble converging minimal hypersurfaces, or H -CMC hypersurfaces, in compact Riemannian manifolds without boundary, of dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>4</m:mn> <m:mo>≤</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>≤</m:mo> <m:mn>7</m:mn> </m:mrow> </m:math> {4\leq n+1\leq 7} . We prove the following upper semicontinuity of index plus nullity: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:munder> <m:mo movablelimits="false">lim sup</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munder> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ind</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>M</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>nul</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>M</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>l</m:mi> </m:munderover> <m:mi>co</m:mi> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>m</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>i</m:mi> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>anl</m:mi> <m:mo>-</m:mo> <m:mi>ind</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>co</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>M</m:mi> <m:mi mathvariant="normal">∞</m:mi> <m:mi>i</m:mi> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>anl</m:mi> <m:mo>-</m:mo> <m:mi>nul</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>co</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>M</m:mi> <m:mi mathvariant="normal">∞</m:mi> <m:mi>i</m:mi> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>J</m:mi> </m:munderover> <m:mi>ind</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="normal">Σ</m:mi> <m:mi>j</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mi>nul</m:mi> <m:msub> <m:mi>ω</m:mi> <m:mrow> <m:msup> <m:mi mathvariant="normal">Σ</m:mi> <m:mi>j</m:mi> </m:msup> <m:mo>,</m:mo> <m:mi>R</m:mi> </m:mrow> </m:msub> </m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="normal">Σ</m:mi> <m:mi>j</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2024-