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Let $\Sigma^n$ be a compact, embedded, oriented hypersurface in a compactoriented Riemannian manifold $N^{n+1}$ with the second fundamental form $h$.Let $H={\rm tr}_{g_{\Sigma}} h$ and $S=|h|^2$ be the mean curvature and squaredlength of the second fundamental form $h$ of $\Sigma$, respectively. If theRicci curvature of $N^{}$ is bounded from below by a positive constant $k>0$and the sectional curvature of $N^{}$ is bounded from above by a positiveconstant $K>0$, then the first nonzero eigenvalue of the Laplacian on $\Sigma$has a lower bound  $$\lambda_1(\Sigma)\geq  \frac {k}{2}-  \frac{H_{\Sigma}}{2}  \left(  nK+\frac{2\sqrt{S_{\Sigma}K}}{\arctan(\sqrt{K})}+2n\sqrt{S_{\Sigma}}+\frac{n}{n+1}H_{\Sigma}  \right).$$ where $H_{\Sigma}=\max_{\Sigma}|H|$ and$S_{\Sigma}=\max_{\Sigma}S$. It extends the result of Choi and Wang [J. Diff.Geom. \textbf{18} (1983), 559--562.] to non-minimal case.

A first eigenvalue estimate for embedded hypersurfaces in positive Ricci curvature manifolds