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Log-scale equidistribution of nodal sets in Grauert tubes Chang, Robert


Let $M$ be a compact real analytic negatively curved manifold. It admits a complexification in which the metric induces a pluri-subharmonic function $\sqrt{\rho}$ whose sublevel sets are strictly pseudo-convex domains $M_\tau$, known as Grauert tubes. The Laplace eigenfunctions on $M$ analytically continue to the Grauert tubes, and their complex nodal sets are complex hypersurfaces in $M_\tau$. Zelditch proved that the normalized currents of integration over the complex nodal sets tend to a single weak limit $dd^c\sqrt{\rho}$ 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 $M_\tau \setminus M$. 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.

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