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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05190-1

Published electronically:
March 16, 2011

MathSciNet review:
2792976

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider a sequence of closed, orientable surfaces of fixed genus in a Riemannian manifold 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 , together with bounds on the geometry of , give an upper bound on the diameter of .

Our proof is modelled on Gromov's compactness theorem for -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:
https://doi.org/10.1090/S0002-9947-2011-05190-1

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.