Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I

TL;DR

This paper extends pluri-potential theory to Grauert tubes of real analytic Riemannian manifolds using eigenfunction continuations, leading to new Weyl laws, zero distributions, and eigenfunction estimates in the complex domain.

Contribution

It introduces a novel framework for pluri-potential theory on Grauert tubes utilizing eigenfunction continuations, with significant results on Weyl laws and eigenfunction zero distributions.

Findings

Weyl laws in the complex domain

Distribution of complex zeros of eigenfunctions

Estimates of triple products of eigenfunctions

Abstract

We develop analogues for Grauert tubes of real analytic Riemannian manifolds (M,g) of some basic notions of pluri-potential theory, such as the Siciak extremal function. The basic idea is to use analytic continuations of eigenfunctions in place of polynomials or sections of powers of positive line bundles for pluripotential theory. The analytically continued Poisson-wave kernel plays the role of Bergman kernel. The main results are Weyl laws in the complex domain, distribution of complex zeros of eigenfunctions on locally symmetric spaces, and estimates of triple products of eigenfunctions.