Lower Bounds on Nodal Sets of Eigenfunctions via the Heat Flow

Abstract

We study the size of nodal sets of Laplacian eigenfunctions on compact Riemannian manifolds without boundary and recover the currently optimal lower bound by comparing the heat flow of the eigenfunction with that of an artificially constructed diffusion process. The same method should apply to a number of other questions; we use it to prove a sharp result saying that a nodal domain cannot be entirely contained in a small neighborhood of a “reasonably flat” surface and recover an older result of Cheng. The arising concepts can be expected to have many more connections to classical theory and we pose some conjectures in that direction.

Keywords:

• Heat Flow
• Laplacian eigenfunctions
• Nodal sets
• Yau conjecture

2010 Mathematics Subject Classification:

• 35B05
• 35J05