Contractibility of moduli spaces of RCD(0,2)-structures

TL;DR

This paper classifies compact topological spaces that admit RCD(0,2)-structures and proves that their moduli spaces are contractible, enhancing understanding of the geometric and topological properties of these non-smooth spaces.

Contribution

It provides a complete classification of spaces with RCD(0,2)-structures and demonstrates that their moduli spaces are contractible, a novel topological insight.

Findings

All moduli spaces of RCD(0,2)-structures are contractible.

Identified the list of compact topological spaces admitting RCD(0,2)-structures.

Described the structure of the moduli spaces for these spaces.

Abstract

This paper focuses on RCD(0,2)-spaces, which can be thought of as possibly non-smooth metric measure spaces with non-negative Ricci curvature and dimension less than 2. First, we establish a list of the compact topological spaces admitting an RCD(0,2)-structure. Then, we describe the associated moduli space of RCD(0,2)-structures for each of them. In particular, we show that all these moduli spaces are contractible.