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 $$\sum _{i=0}^n{b}^i(\Sigma )\le C_A(1+index(\Sigma ))\text{ if 3≤n+1≤7},$$ $${\mathcal{H}}^{n-7}(Sing(\Sigma ))\le C_A(1+index(\Sigma ){)}^{7/n}\text{ if n+1≥8},$$ 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.

arXiv commentv2: Section 4 improved, minor corrections, results unchanged. v3: Corrections suggested by referees, correction suggested by Giada Franz and Santiago Cordero Misteli. To appear in Duke Math. J

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• Generic scarring for minimal hypersurfaces along stable hypersurfaces
• Essential minimal volume of Einstein 4-manifolds