Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Morse index, Betti numbers and singular set of bounded area minimal
  hypersurfaces   We introduce a combinatorial argument to study closed minimal hypersurfaces
of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian
manifold and $\Sigma\subset M$ be a closed embedded minimal hypersurface with
area at most $A>0$ and with a singular set of Hausdorff dimension at most
$n-7$. We show the following bounds: there is $C_A>0$ depending only on $n$,
$g$, and $A$ so that $$\sum_{i=0}^n b^i(\Sigma) \leq C_A
\big(1+index(\Sigma)\big) \quad \text{ if $3\leq n+1\leq 7$},$$
$$\mathcal{H}^{n-7}\big(Sing(\Sigma)\big) \leq C_A
\big(1+index(\Sigma)\big)^{7/n} \quad \text{ if $n+1\geq 8$},$$ where $b^i$
denote the Betti numbers over any field, $\mathcal{H}^{n-7}$ is the
$(n-7)$-dimensional Hausdorff measure and $Sing(\Sigma)$ is the singular set of
$\Sigma$. In fact in dimension $n+1=3$, $C_A$ depends linearly on $A$. We list
some open problems at the end of the paper.

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The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: physics
1
The categories of this paper are: cs
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn