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Strong Kato limit can be branching

Gilles Carron, Ilaria Mondello, David Tewodrose

TL;DR

The paper expands the landscape of metric space limits under weak curvature control by constructing non-collapsed strong Kato limits that are branching and fail CD and MCP conditions, as well as a compact example not obtainable via Petersen–Wei-type $L^p$-Ricci bounds. It develops a conformal-geometry framework showing that mild regularity on the conformal factor preserves strong Kato limits, and provides a two-pronged demonstration: a two-dimensional branching example and a compact obstruction to L^p-based limiting procedures. The results rely on heat-kernel/Kato estimates, two-dimensional convergence theory for surfaces with bounded integral curvature, and Debin/Alexandrov-type compactness. Altogether, the work demonstrates that non-collapsed strong Kato limits can be strictly larger than finite-dimensional $\mathrm{RCD}$ spaces and Petersen–Wei limits, motivating further study of Kato-limit structures in metric geometry.

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 à la Petersen--Wei.

Strong Kato limit can be branching

TL;DR

The paper expands the landscape of metric space limits under weak curvature control by constructing non-collapsed strong Kato limits that are branching and fail CD and MCP conditions, as well as a compact example not obtainable via Petersen–Wei-type -Ricci bounds. It develops a conformal-geometry framework showing that mild regularity on the conformal factor preserves strong Kato limits, and provides a two-pronged demonstration: a two-dimensional branching example and a compact obstruction to L^p-based limiting procedures. The results rely on heat-kernel/Kato estimates, two-dimensional convergence theory for surfaces with bounded integral curvature, and Debin/Alexandrov-type compactness. Altogether, the work demonstrates that non-collapsed strong Kato limits can be strictly larger than finite-dimensional spaces and Petersen–Wei limits, motivating further study of Kato-limit structures in metric geometry.

Abstract

We provide an example of a non-collapsed strong Kato limit that is branching, essentially branching, and satisfies neither the nor the conditions for any and . 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 bound à la Petersen--Wei.

Paper Structure

This paper contains 13 sections, 14 theorems, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Conformal metric with low regularity and strong Kato limits
  3. The Kato constant of a smooth conformal metric
  4. Conformal metric with low regularity
  5. Proof of Theorem \ref{['thm:exA']}
  6. The example is a non-collapsed strong Kato limit
  7. The example is branching
  8. The example is essentially branching and not CD
  9. The example is degenerate and not finite-dimensional MCP
  10. Proof of Theorem \ref{['prop:NotLp']}
  11. Strong Kato and $L^p$ bounds.
  12. Convergence of surfaces with Bounded Integral Curvature.
  13. Proof of Theorem \ref{['prop:NotLp']}

Key Result

Theorem 1

The surface $(\mathcal{C},\widehat{d} )$ is a branching non-collapsed strong Kato limit. Moreover, the metric measure space $(\mathcal{C},\widehat{\mathop{\mathrm{d}}\nolimits},\widehat{\mathcal{H}}^2)$ satisfies neither the $\mathop{\mathrm{CD}}\nolimits(K,\infty)$ nor the $\mathop{\mathrm{MCP}}\no

Figures (4)

  • Figure 1: Branching geodesics.
  • Figure 2: Branching set of geodesics between $A_0$ and $A_1$.
  • Figure 3: Negligible set of $t$-intermediary points $\gamma(t)$ for $t$ close to $1$.
  • Figure 4: The space $(\mathbb{S}^2,\overline{g})$.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 18 more