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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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
  • MR Author ID: 1008414
  • Email: b.g.sharp@leeds.ac.uk
  • Received by editor(s): September 15, 2016
  • Received by editor(s) in revised form: December 22, 2016
  • Published electronically: February 26, 2018
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 4373-4399
  • MSC (2010): Primary 53A10; Secondary 49Q05, 58E12
  • DOI: https://doi.org/10.1090/tran/7168
  • MathSciNet review: 3811532