> [!definition] > > Let $X$ be a smooth [[Compactness|compact]] and [[Connected|connected]] [[Riemannian Manifold|Riemannian]] [[n-Manifold|n-manifold]], and $dV$ be the [[Radon Measure|measure]] [[Measure Associated With Form|induced by]] the [[Riemannian Volume Form|Riemannian volume form]]. If $\Delta$ is the [[Laplacian]], then $\lambda \in \real$ is a solution to the closed eigenvalue problem if there exists $\phi \in C^2(X)$ such that > $ > \Delta \phi + \lambda \phi = 0 > $ > [!theorem] > > 1. All solutions are non-negative. > 2. Each solution has finite geometric multiplicity. > 3. Distinct eigenspaces are orthogonal. > 4. All sequences of solutions accumulate at $\infty$. > > *Proof*. Let $\phi \in C^2(X)$ such that $\Delta \phi + \lambda \phi = 0$, then by [[Green's Formulas]], > $ > \begin{align*} > -\lambda \norm{f}_2^2 &= \int_X f\Delta f dV \\ > &= -\int_X \angles{\grad f, \grad{f}}dV \\ > \lambda &= \frac{\norm{\grad{f}}_2^2}{\norm{f}_2^2} > \end{align*} > $