LOCAL VERSION OF COURANT’S NODAL DOMAIN THEOREM

Let (Mⁿ, g) be a compact n-dimensional Riemannian manifold without boundary, where g = (g_ij) is C¹-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 Volume}^Volume_g^g₍⁽_M^B)₎k + C₂k^{n− n1} , where C₁, C₂ 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.