Strong Kato limit can be branching

TL;DR

This paper constructs examples of non-collapsed strong Kato limits that are branching and do not satisfy common curvature-dimension conditions, showing new phenomena in metric measure geometry.

Contribution

It provides the first examples of branching strong Kato limits that violate standard curvature conditions, expanding understanding of limit spaces.

Findings

Existence of non-collapsed strong Kato limits that are branching.

Such spaces do not satisfy $ ext{CD}(K, ext{infty})$ or $ ext{MCP}(K,N)$ conditions.

Construction of a compact example not obtainable as Gromov-Hausdorff limit of certain Riemannian surfaces.

Abstract

We provide an example of a non-collapsed strong Kato limit that is branching, essentially branching, and satisfies neither the $\mathrm{CD}(K,\infty)$ nor the $\mathrm{MCP}(K,N)$ conditions for any $K \in \mathbb{R}$ and $N \in [1,+\infty)$. In particular, this space is not a Ricci limit space. We also construct a compact non-collapsed strong Kato limit that cannot be obtained as Gromov-Hausdorff limit of closed Riemannian surfaces satisfying a uniform small $L^p$ bound \`a la Petersen--Wei.