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Journal Article

#### Curvature estimates for surfaces with bounded mean curvature

##### Locator

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##### Fulltext (public)

1007.3425

(Preprint), 162KB

1007.3425v3.pdf

(Preprint), 170KB

TransAMS-2012-05487-0.pdf

(Any fulltext), 243KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Bourni, T., & Tinaglia, G. (2012). Curvature estimates for surfaces with bounded
mean curvature.* Transactions of the American Mathematical Society,* *364*(11
), 5813-5828. Retrieved from http://arxiv.org/abs/1007.3425.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0012-C6CB-F

##### Abstract

Estimates for the norm of the second fundamental form, $|A|$, play a crucial
role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the
surface cannot bend too sharply. In this paper we prove that for an embedded
geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior
points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently
small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for
minimal surfaces.