A Bernstein type of inequality for eigenfunctions (Q1907676)

Let \(M\) be a smooth closed Riemannian manifold of dimension \(n\). \textit{H. Donnelly} and \textit{C. Fefferman} proved: Theorem. Let \(u\) be an eigenfunction of the Laplacian with eigenvalue \(\lambda\). Then \[ \max_{B_r(x)} |\nabla u|\leq C_1 \lambda^{(n + 2)/4} r^{-1} \max_{B_r(x)} |u|. \] They conjectured that the factor \(\lambda^{(n + 2)/4}\) could be replaced by \(\lambda^{1/2}\). The author establishes this estimate in the special case of surfaces \((n = 2)\) and small radii: Theorem. Let \(u\) be an eigenfunction of the Laplacian with eigenvalue \(\lambda\) and \(n = 2\). Let \(r \leq C_2 \lambda^{-1/4}\). Then \[ \max_{B_r(x)} |\nabla u|\leq C_3 \lambda^{1/2} r^{-1} \max_{B_r(x)} |u|. \] Here the constants \(c_i = c_i(H, D)\) depend only upon an upper bound \(H\) of the absolute value of the sectional curvature and on the diameter \(D\) of the manifold. This inequality is a generalized Bernstein inequality.