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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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Surfaces of bounded mean curvature in Riemannian manifolds
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by Siddhartha Gadgil and Harish Seshadri PDF
Trans. Amer. Math. Soc. 363 (2011), 3977-4005 Request permission

Abstract:

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two.

As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface $F\subset M$, together with bounds on the geometry of $M$, give an upper bound on the diameter of $F$.

Our proof is modelled on Gromov’s compactness theorem for $J$-holomorphic curves.

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Additional Information
  • Siddhartha Gadgil
  • Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
  • Email: gadgil@math.iisc.ernet.in
  • Harish Seshadri
  • Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
  • MR Author ID: 712201
  • Email: harish@math.iisc.ernet.in
  • Received by editor(s): March 16, 2009
  • Published electronically: March 16, 2011
  • Additional Notes: This work was supported by the University Grants Commission (UGC). The second author was supported by DST Grant No. SR/S4/MS-283/05
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 3977-4005
  • MSC (2010): Primary 53C21
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05190-1
  • MathSciNet review: 2792976