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 (1983) and prove that any Laplace nodal domain ${\Omega }_{\lambda }\subseteq M$ almost fully contains a ball of radius $\sim 1∕\sqrt{{\lambda }_1\left({\Omega }_{\lambda }\right)}$, and such a ball can be centred at any point of maximum of the Dirichlet ground state ${\phi }_{{\lambda }_1\left({\Omega }_{\lambda }\right)}$. This also gives a slight refinement of a result by Mangoubi (2008) concerning the inradius of nodal domains. On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside $M$.