First eigenvalue of embedded minimal surfaces in $S^3$

Abstract

We prove that for an embedded minimal surface $\Sigma$ in $S^3$, the first eigenvalue of the Laplacian operator ${\lambda }_1$ satisfies ${\lambda }_1\ge 1+{\epsilon }_g$, where ${\epsilon }_g>0$ is a constant depending only on the genus $g$ of $\Sigma$. This improves previous result of Choi-Wang.