Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

Abstract

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is quantised in terms of the total curvature of some limit surface, plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in Euclidean space - all of which are finite. Thus, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary.

27 pages; v3: final version to be published in Trans. Amer. Math. Soc

References in corpus (2)

• Minimal hypersurfaces with bounded index
• On the topology and index of minimal surfaces

Cited by in corpus (7)

• Minimal hypersurfaces with bounded index
• On the topology and index of minimal surfaces
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• Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces
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• Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers
• Bubbling analysis and geometric convergence results for free boundary minimal surfaces