Calculus of Variations and Geometric Measure Theory

R. Buzano - B. Sharp

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

created by muller on 12 Jun 2018
modified on 13 Jun 2022


Published Paper

Inserted: 12 jun 2018
Last Updated: 13 jun 2022

Journal: Transactions of the American Mathematical Society
Volume: 370
Year: 2018
Doi: 10.1090/tran/7168

ArXiv: 1512.01047 PDF


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.