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

TL;DR

This paper refines volume estimates to derive a sharp inequality relating the genus of embedded surfaces in the 3-sphere to curvature and second fundamental form, leading to compactness results for minimal surfaces with bounded curvature.

Contribution

It introduces a new sharp pinching estimate for the genus of surfaces in the 3-sphere involving integral curvature terms, improving understanding of embedded surface geometry.

Findings

Derived a sharp genus estimate involving integral of traceless second fundamental form.

Proved compactness of minimal surfaces with bounded curvature in the $C^k$ topology.

Established a new inequality connecting genus, volume, and curvature for surfaces in manifolds.

Abstract

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 $k\ge2$.