Nodal geometry on Riemannian manifolds (Q803629)

Let M be a compact connected Riemannian manifold of dimension m without boundary. Let u be an eigenfunction of the Laplacian corresponding to an eigenvalue \(\lambda >1\). The authors show: Theorem: (a) \(\| \log | u| \|_{BMO}\leq c(M)\lambda^ m\cdot \log (\lambda).\) (b) Let B be a ball in M and let \(\Omega\subset B\) be one of the connected components of \(\{\) \(x\in B:\) u(x)\(\neq 0\}\). If \(\Omega\) intersects the middle half of B, \[ vol(\Omega)\geq c(M)\lambda^{-2m^ 2-m/2}(\log \lambda)^{-2m}vol(B). \] Remark: Similar theorems have been proved by H. Donnelly and C. Fefferman with \(\lambda^ m\cdot \log (\lambda)\) replaced by \(\lambda^{m(m+2)/4}\) in (a) and with the second bound replaced by \(\lambda^{-(m+m^ 2(m+2))/2}\) in (b).