Gadgil, Siddhartha and Seshadri, Harish
(2011)
*Surfaces of Bounded Mean Curvature In Riemannian Manifolds.*
In: Transactions of the American Mathematical Society, 363
(8).
pp. 3977-4005.

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## 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 of 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.

Item Type: | Journal Article |
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Publication: | Transactions of the American Mathematical Society |

Publisher: | American Mathematical Society |

Additional Information: | Copyright of this article belongs to American Mathematical Society. |

Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |

Date Deposited: | 03 Aug 2011 05:57 |

Last Modified: | 03 Aug 2011 05:57 |

URI: | http://eprints.iisc.ac.in/id/eprint/39594 |

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