Let $M^n$ be a minimal hypersurface in $R^{n+1}$ with finite index. Can $\lambda_1(M^n)$ be a positive number? Are there examples of such hypersurfaces? (Even surfaces...)
Recall that $M^n$ has finite index if
$$\int_{M^n \setminus D} |A|^2 \phi^2 \leq \int_{M^n \setminus D} |\nabla \phi|^2$$
for all $\phi \in C^\infty_0 (M^n\setminus D)$, where $D$ is a compact domain and $A$ is the second fundamental form of $M^n$, and the first eigenvalue of the Laplacian is
$\lambda_1(M^n)=\inf_{\phi \in C^\infty_0 (M^n)} \frac{\int_{M^n} |\nabla \phi |^2 }{\int_{M^n} \phi^2}.$
Thank you!
