1

Let $\left(M^n, g\right)$ be a closed Riemannian manifold with $C^1$-smooth $g_{i j}$. The spectrum of the Laplace operator on $M$ is discrete. There is a sequence of eigenvalues

$$
0=\lambda_1<\lambda_2 \leq \lambda_3 \ldots
$$

that tend to $\infty$ and a sequence of (real) eigenfunctions $u_k$ such that

$$
\Delta_g u_k+\lambda_k u_k=0 .
$$

Our enumeration of eigenvalues is non-standard. We start with $\lambda_1=0$ and $u_1=1$ on $M$. The nodal domains of $u_k$ are the connected components of $M \backslash Z_{u_k}$, where $Z_{u_k}$ is the zero set of $u_k\left(Z_{u_k}\right.$ is called the nodal set of $\left.u_k\right)$. The Courant nodal domain theorem states that the $k$-th eigenfunction $u_k$ has at most $k$ nodal domains. If the multiplicity of an eigenvalue is more than 1 , one may enumerate the eigenfunctions corresponding to this eigenvalue in any order. Our main result is the local version of Courant’s theorem.