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

Buzano, Reto; Sharp, Ben

doi:10.1090/tran/7168

Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area
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by Reto Buzano and Ben Sharp PDF

Trans. Amer. Math. Soc. 370 (2018), 4373-4399 Request permission

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 hypersurface, 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.

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