MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.