Neumann Eigenvalue is Nonnegative

Theorem

Let $\struct {M, g}$ be a compact connected Riemannian manifold with non-empty boundary $\partial M$.

Let $\lambda$ be a Neumann eigenvalue of $M$.

Then $\lambda$ is non-negative.

That is, $0$ is a Neumann eigenvalue, and all the other Neumann eigenvalues are strictly positive.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems