Curvature estimates for surfaces with bounded mean curvature

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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 L² norm of |A|, |A| is bounded at interior points, provided that the W¹,ᵖ norm of its mean curvature is sufficiently small, p>2. In doing this we generalize some renowned estimates on |A| for minimal surfaces.