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

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.

First in a series. Much of the article is expository. In particular, it goes over Hadamard's parametrix construction for his branched meromorphic fundamental solution and applies it to construct a parametrix for the Poisson wave group

References in corpus (1)

• A Haar component for quantum limits on locally symmetric spaces

Cited by in corpus (12)

• Measure of nodal sets of analytic Steklov eigenfunctions
• Renormalization of quantum field theory on curved space-times, a causal approach
• Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
• Geometric wave propagator on Riemannian manifolds
• Eigenfunctions and Nodal Sets
• Dynamical residues of Lorentzian spectral zeta functions
• Complex structures adapted to magnetic flows
• Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces
• Asymptotic properties of Bergman kernels for potentials with Gevrey regularity
• Algebras of pseudo-differential operators acting on holomorphic Sobolev spaces
• $L^{\infty }$ norms of Husimi distributions of eigenfunctions
• Scaling asymptotics for Szegő kernels on Grauert tubes