  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
$\Sigma$ in a $3$-dimensional orientable Riemannian manifold $M$ whose
sectional curvature is bounded below by $1$, then $4 \pi^{2} g(\Sigma) \le
2\left(2 \pi^{2}-|M|\right)+\int_{\Sigma} f(|\stackrel \circ A|)$, where $
\stackrel \circ A $ 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 $\|A\|_{L^3(\Sigma)}$ is compact in
the $C^k$ topology for any $k\ge2$.
