Surfaces of Bounded Mean Curvature In Riemannian Manifolds

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
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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