# Minimal hypersurfaces with finite index and eigenvalues of the Laplacian

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!

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