The first eigenvalue of embedded minimal hypersurfaces in the unit sphere

In this article, we prove that for an embedded minimal hypersurface ${\mathrm{\Sigma }}^m$ in $S^{m+1}$, the first eigenvalue ${\lambda }_1$ of the Laplacian operator on $\mathrm{\Sigma }$ satisfies: $${\lambda }_1>\frac{m}{2}+G(m,\mathrm{\mid }A{\mathrm{\mid }}_{\max },\mathrm{\mid }A{\mathrm{\mid }}_{\min }),$$ where $\mathrm{\mid }A{\mathrm{\mid }}_{\max }$ and $\mathrm{\mid }A{\mathrm{\mid }}_{\min }$ denote the maximum and minimum of the norm of the second fundamental form on $\mathrm{\Sigma }$, respectively; $G(m,\mathrm{\mid }A{\mathrm{\mid }}_{\max },\mathrm{\mid }A{\mathrm{\mid }}_{\min })$ is a positive constant that depends only on $m,\mathrm{\mid }A{\mathrm{\mid }}_{\max },\mathrm{\mid }A{\mathrm{\mid }}_{\min }$. In particular, when the norm $\mathrm{\mid }A\mathrm{\mid }$ of the second fundamental form is constant, we can obtain a gap depending only on $m$, i.e., $${\lambda }_1>(\frac{1}{2}+c)m,$$ where $c$ is a positive absolute constant. This improves Choi and Wang’s previous result \cite{chw1983first} that ${\lambda }_1\ge \frac{m}{2}$. Our result shows that one can improve Choi and Wang’s result directly without proving Chern’s conjecture. This also generalizes Tang and Yan’s work \cite{tangyan2013isoparametric}. Based on the proof of the result above, using the lower bound of the first Steklov eigenvalue, we prove that if the norm $\mathrm{\mid }A\mathrm{\mid }$ of the second fundamental form is constant, then $$\mathrm{\mid }A\mathrm{\mid }\le \frac{C(m)\text{Volume}(\mathrm{\Sigma })}{\text{Volume}(S^m)},$$ where $C(m)$ is a constant that depends only on $m$. This provides a uniform estimate for the scalar curvature of embedded minimal hypersurfaces with constant norm of the second fundamental form. Moreover, this may be useful for Chern’s problem.