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 $\Sigma$ 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 $k\ge2$.