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

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

2011-07-03

arxiv.org/abs/1107.0463v1

Mathematics

Spectral Theory

First in a series. Much of the article is expository. In particular, it goes over Hadamard's parametrix construction for his b

Scientific paper

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.

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