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Log-scale equidistribution of nodal sets in Grauert tubes

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Log-scale equidistribution of nodal sets in Grauert tubes
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20
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Identifikatoren
Herausgeber
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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.