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# Mathematical Research Letters

## Volume 18 (2011)

### Number 2

### A geometric covering lemma and nodal sets of eigenfunctions

Pages: 337 – 352

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n2.a11

#### Authors

#### Abstract

The main purpose of this paper is two-fold. On one hand, we prove a sharper covering lemma in Euclidean space $\mathbb R^n$ for all $n\ge2$ (see Theorem \ref{newcovering}). On the other hand, we apply this covering lemma to improve existing results for BMO and volume estimates of nodal sets for eigenfunctions $u$ satisfying $\bigtriangleup u+\lambda u=0$ on $n$-dimensional Riemannian manifolds when $\lambda$ is large (see Theorems \ref{BMO1}, \ref{volume}). We also improve the BMO estimates for the function $q=|\nabla u|^2+\frac{\lambda}{n}u^2$ (see Theorem \ref{BMO2}). Our covering lemma sharpens substantially earlier results and is fairly close to the optimal one we can expect (Conjecture \ref{conjecture}).