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

Abstract

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.