Some applications of heat flow to Laplace eigenfunctions

Abstract

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

Keywords:

• avoided crossings
• Brownian motion
• heat equation
• Laplace eigenfunction
• mass concentration
• nodal geometry

Acknowledgements

The work was started when the second author was at the Technion, Israel supported by the Israeli Higher Council and the research of the second author leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 637851). The work was continued when the second author was at the Courant Institute of Mathematical Sciences hosted by Fanghua Lin, the Max Planck Institute for Mathematics, Bonn and finished when he was at IIT Bombay. He wishes to deeply thank them all. The first author gratefully acknowledges the Max Planck Institute for Mathematics, Bonn and the Fraunhofer Institute, IAIS, for providing ideal working conditions. The first author’s research is also within the scope of the ML2R project and Research Center, ML at Fraunhofer, IAIS. The authors also thank Steve Zelditch, Michiel van den Berg, Gopal Krishna Srinivasan, Stefan Steinerberger and Rajat Subhra Hazra for helpful conversations. In particular the authors are deeply grateful to Emanuel Milman for reading a draft version in detail and pointing out several corrections and improvements. Thanks are also due to the two anonymous Referees whose comments and suggestions led to several significant improvements in the presentation.

Notes

1 There are two popular proofs of this fact, one uses domain monotonicity of Dirichlet eigenvalues, and the other uses Harnack inequality on the harmonic function $e^{\sqrt{\lambda }t}{\phi }_{\lambda }(x)$ in $\mathbb{R}\times M.$