Some applications of heat flow to Laplace eigenfunctions

We consider mass concentration properties of Laplace eigenfunctions φλ, that is, smooth functions satisfying the equation −Δφλ=λφλ, on a closed Riemannian manifold. Using heat diffusion technique, we first discuss concentration/localization around their nodal sets. Second, problem avoided crossings and (non)existence domains which continue to be thin over relatively long distances. Further, using above techniques, decay Euclidean have central “thick” part “thin” elongated branches representing tunnels sub-wavelength opening. Finally, in an Appendix, record some new observations regarding sub-level sets interactions different level