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 (Mn+1,g) be a closed Riemannian manifold, and let Σ⊂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 CA>0 depending only on n, g, and A so that ∑i=0nbi(Σ)≤CA(1+index(Σ))if3≤n+1≤7,Hn−7(Sing(Σ))≤CA(1+index(Σ))7∕nifn+1≥8, where bi denote the Betti numbers over any field, Hn−7 is the (n−7)-dimensional Hausdorff measure, and Sing(Σ) is the singular set of Σ. In fact, in dimension n+1=3, CA depends linearly on A. We list some open problems at the end of the paper.

Topics & Concepts