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

Abstract By refining the volume estimate of Heintze and Karcher 11, we obtain a sharp pinching estimate for the genus of a surface in S^3 S 3, 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 4 ^2 g () 2 (2 ^2-|M|) + f (| {A ^ }|) 4 π 2 g (Σ) ≤ 2 2 π 2 - | M | + ∫ Σ f (| A ∘ |), where {A ^ } 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 ₋℃ () ‖ A ‖ L 3 (Σ) is compact in the Cᵏ C k topology for any k 2 k ≥ 2.