An estimate for the genus of embedded surfaces in the 3-sphere

By refining the volume estimate of Heintze and Karcher \cite{HK}, 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 $\mathrm{\Sigma }$ in a $3$-dimensional orientable Riemannian manifold $M$ whose sectional curvature is bounded below by $1$, then $4{\pi }^{2}g(\mathrm{\Sigma })\le 2(2{\pi }^2-\mathrm{\mid }M\mathrm{\mid })+{\int }_{\mathrm{\Sigma }}f(\mathrm{\mid }\mathrm{\mid })$, where $$ is the traceless second fundamental form and $f$ is an explicit function. As a result, the space of closed orientable embedded minimal surfaces $\mathrm{\Sigma }$ with uniformly bounded $\mathrm{\parallel }A{\mathrm{\parallel }}_{L^3(\mathrm{\Sigma })}$ is compact in the $C^k$ topology for any $k\ge 2$.