Sign and area in nodal geometry of Laplace eigenfunctions
By Fedor Nazarov, L. Polterovich, M. Sodin. 2004
58 citations recorded by the source corpus.
The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign( f ) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line { f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/421a3e21fc5cbe1ecf5212f319e78aec4218f397
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three (2016, methodology): “This well-known trick was successfully exploited for example in [8, 14, 11].”
- Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure (2016, background): “In dimension two, understanding of nodal sets of harmonic functions is better due to complex analysis techniques and topological reasons; see [18].”
- Local Asymmetry and the Inner Radius of Nodal Domains (2007, methodology): “In [10], the authors indicate how to obtain a local asymmetry estimate for harmonic functions in dimensions n ≥ 3 based on Carleman’s method.”
- Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory (2004, background): “This is related to so-called doubling constant of φi (see [NPS]).”