English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce a combinatorial argument to study closed minimal hypersurfacesof bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannianmanifold and $\Sigma\subset M$ be a closed embedded minimal hypersurface witharea 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 listsome open problems at the end of the paper.

Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces