Let \((M,g)\) be an analytic Riemannian manifold with a analytic metric \(g\). Then every smooth eigenfunction of the geometric Laplacian is analytic .
Proof
Let \(p\in M\) and \((U,\varphi)\) an analytic chart around \(p\). Then in local coordinates we have
\begin{equation*} L\hat{u} = - \frac{1}{\sqrt{\det g} } \frac{\partial }{\partial x^i}\biggl(g^{ij} \sqrt{\det g} \frac{\partial \hat{u}}{\partial x^j}\biggr) - \lambda \hat{u}=0, \qquad \text{in } U \end{equation*}where \(\hat{u}\) is the coordinate representation of \(u\). Since \(g\) is positive definite according to the definition, \(L\) is an elliptic operator (indeed it is uniformly elliptic). All coefficients of \(L\) are analytic because of the analyticity of \(g\) and therefore we may apply (0x67504ba2) implying that \(\hat{u}\) is analytic.