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


Authors: Reto Buzano and Ben Sharp
Journal: Trans. Amer. Math. Soc. 370 (2018), 4373-4399
MSC (2010): Primary 53A10; Secondary 49Q05, 58E12
DOI: https://doi.org/10.1090/tran/7168
Published electronically: February 26, 2018
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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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Additional Information

Reto Buzano
Affiliation: School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
Email: r.buzano@qmul.ac.uk

Ben Sharp
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Address at time of publication: School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Email: b.g.sharp@leeds.ac.uk

DOI: https://doi.org/10.1090/tran/7168
Received by editor(s): September 15, 2016
Received by editor(s) in revised form: December 22, 2016
Published electronically: February 26, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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