Lower bounds on nodal sets of eigenfunctions via the heat flow

摘要

We study the size of nodal sets of Laplacian eigenfunctions on compactRiemannian manifolds without boundary and recover the currently optimal lowerbound by comparing the heat flow of the eigenfunction with that of anartifically constructed diffusion process. The same method should apply to anumber of other questions; for example, we prove a sharp result saying that anodal domain cannot be entirely contained in a small neighbourhood of a'reasonably flat' surface. We expect the arising concepts to have moreconnections to classical theory and pose some conjectures in that direction.