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 let $\mathrm{\Sigma }\subset M$ 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

$$\begin{array}{rcl}\sum _{i=0}^n{b}^i(\mathrm{\Sigma })& \le & C_A(1+index(\mathrm{\Sigma }))\text{if}3\le n+1\le 7,\\ {\mathcal{H}}^{n-7}(Sing(\mathrm{\Sigma }))& \le & C_A{(1+index(\mathrm{\Sigma }))}^{7∕n}\text{if}n+1\ge 8,\end{array}$$

where $b^i$ denote the Betti numbers over any field, ${\mathcal{H}}^{n-7}$ is the $(n-7)$-dimensional Hausdorff measure, and $Sing(\mathrm{\Sigma })$ is the singular set of Σ. In fact, in dimension $n+1=3$, $C_A$ depends linearly on A. We list some open problems at the end of the paper.