Bottom of spectrum of Kahler manifolds with strongly pseudoconvex boundary

TL;DR

This paper investigates the spectral properties of complete Kahler manifolds with strongly pseudoconvex boundaries, exploring the relationship between their geometric structure and boundary CR geometry, and providing partial results towards a conjecture.

Contribution

It formulates a conjecture linking the bottom of the spectrum of Kahler-Einstein manifolds to boundary CR geometry and offers partial proofs.

Findings

Partial results supporting the conjecture

Analysis of asymptotic geometry of the manifolds

Formulation of a spectral-boundary relationship

Abstract

We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.