Nodal Geometry, Heat Diffusion and Brownian Motion

Abstract

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain ${\Omega }_{\lambda }$ almost fully contains a ball of radius $\sim \frac{1}{\sqrt{\lambda }}$. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other hand, we also prove that no nodal domain can be contained in a reasonably thin tubular neighbourhood of unions of finitely many surfaces inside $M$.

16 pages, final accepted version, comments welcome!

References in corpus (2)

• Local Asymmetry and the Inner Radius of Nodal Domains
• Tubular Neighborhoods of Nodal Sets and Diophantine Approximation

Cited by in corpus (1)

• Density of zero sets for sums of eigenfunctions