Local version of Courant’s nodal domain theorem

Abstract

$\def\Volume{\operatorname{Volume}}$Let $(M^n, g)$ be a compact $n$-dimensional Riemannian manifold without boundary, where $g = (g_{ij})$ is $C^1$-smooth. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on $M$. Let $B$ be a ball on $M$. We prove that the number of nodal domains of $u_k$ that intersect $B$ is not greater than\[C_1 \dfrac{\Volume_g (B)}{\Volume_g(M)} k + C_2 k^\frac{n-1}{n} \: ,\]where $C_1, C_2$ depend on $M$. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.