A Sharp Estimate for the Genus of Embedded Surfaces in the 3-Sphere

By refining the volume estimate of Heintze and Karcher [11], we obtain a sharp pinching estimate for the genus of a surface in S3, which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if g is the genus of a closed orientable surface Σ in a 3-dimensional orientable Riemannian manifold M whose sectional curvature is bounded below by 1, then 4π2g(Σ)≤22π2-|M|+∫Σf(|A∘|), where A∘ is the traceless second fundamental form and f is an explicit function. As a result, the space of closed orientable embedded minimal surfaces Σ with uniformly bounded ‖A‖L3(Σ) is compact in the Ck topology for any k≥2.