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 \(\mathbb S^{3}\), 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 \(\Sigma \) in a 3-dimensional orientable Riemannian manifold M whose sectional curvature is bounded below by 1, then \(4 \pi ^{2} g(\Sigma ) \le 2\left( 2 \pi ^{2}-|M|\right) +\int _{\Sigma } f(|{\mathop {A}\limits ^{\circ }}|)\), where \( {\mathop {A}\limits ^{\circ }} \) is the traceless second fundamental form and f is an explicit function. As a result, the space of closed orientable embedded minimal surfaces \(\Sigma \) with uniformly bounded \(\Vert A\Vert _{L^3(\Sigma )}\) is compact in the \(C^k\) topology for any \(k\ge 2\).