Tubular Neighborhoods of Nodal Sets and Diophantine Approximation

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.Comment: 22 pages; revised version containing full proof of lower bound; reference added; to appear in Amer. J. Math

Scalar curvature and QQ-curvature of random metrics

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.Comment: The proof of Proposition 3.10 has been correcte