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 Σ 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 Σᵢ₌₀ⁿ bᶦ(Σ) ≤ C_A (1 + index(Σ)) if 3 ≤ n + 1 ≤ 7, Hⁿ⁻⁷(Sing(Σ)) ≤ C_A (1 + index(Σ))^(7/n) if n + 1 ≥ 8, where bᶦ 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, C_A depends linearly on A. We list some open problems at the end of the paper.