We're sorry but this page doesn't work properly without JavaScript enabled. Please enable it to continue.
Feedback

Log-scale equidistribution of nodal sets in Grauert tubes

Formal Metadata

Title
Log-scale equidistribution of nodal sets in Grauert tubes
Title of Series
Number of Parts
20
Author
License
CC Attribution - NonCommercial - NoDerivatives 4.0 International:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Identifiers
Publisher
Release Date
Language

Content Metadata

Subject Area
Genre
Abstract
Let M be a compact real analytic negatively curved manifold. It admits a complexification in which the metric induces a pluri-subharmonic function ρ−−√ whose sublevel sets are strictly pseudo-convex domains Mτ, known as Grauert tubes. The Laplace eigenfunctions on M analytically continue to the Grauert tubes, and their complex nodal sets are complex hypersurfaces in Mτ. Zelditch proved that the normalized currents of integration over the complex nodal sets tend to a single weak limit ddcρ−−√ along a density one subsequence of eigenvalues. In this talk, we discuss a joint work with Steve Zelditch, in which we show that the weak convergence result holds `on small scale,' namely, on logarithmically shrinking Kaehler balls whose centers lie in Mτ∖M. The main technique is a Poisson-FBI transform relating QE on Kaehler balls to QE on the real domain. Similar small-scale QE results were obtained in the Riemannian setting by Hezari-Riviere and Han, and in the ample line bundle setting by Chang-Zelditch.