## 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