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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Surfaces of bounded mean curvature in Riemannian manifolds


Authors: Siddhartha Gadgil and Harish Seshadri
Journal: Trans. Amer. Math. Soc. 363 (2011), 3977-4005
MSC (2010): Primary 53C21
Published electronically: March 16, 2011
MathSciNet review: 2792976
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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 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
Email: harish@math.iisc.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05190-1
PII: S 0002-9947(2011)05190-1
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.