Saturation and isomorphism of abstract harmonic spaces

TL;DR

This paper models abstract harmonic spaces using continuous first-order logic, characterizes low-dimensional manifolds by U-rank and saturation, and explores measure-theoretic and topological properties.

Contribution

It introduces a logical framework for harmonic spaces, characterizes certain manifolds via model-theoretic properties, and connects harmonic measures with Keisler measures.

Findings

Characterization of $M^n$ ($n extless= 2$) by U-rank and saturation.

Polar sets characterized by omitting type theorem conditions.

Bijective correspondence between harmonic measures and Keisler measures.

Abstract

This paper models the theory of abstract harmonic spaces in the syntax of the continuous first-order logic of Banach lattices. It addresses a topological question asking when a one-to-one harmonic map onto smooth manifolds $M^n$ is a diffeomorphism. We give $M^n$ ($n\le 2$) a characterization by $U$-rank and elementary saturation for large cardinals. Polar sets are characterized by several equivalent conditions from the omitting type theorem. Consequently, harmonic measures on the ideal boundary in Martin representation are bijectively mapped to Keisler measures supported on non-principal types. Further problems concerning o-minimality and non-local potentials are finally discussed.