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From almost smooth spaces to RCD spaces

Shouhei Honda, Song Sun

TL;DR

The paper addresses how to locally characterize $ ext{RCD}(K,N)$ spaces on almost smooth metric measure spaces under a uniformly local PI framework, introducing the notions of SL and QL to connect local geometry with global curvature-dimension bounds. It proves two sharp equivalences: (i) an almost smooth space is $ ext{RCD}(K,N)$ iff it is PI with SL and QL and the $N$-Bakry-Émery Ricci curvature bound holds on the smooth part, and (ii) if the singular set has codimension at least $4$, the QL assumption can be dropped and the same characterization holds; these results extend to non-smooth frameworks. The work also applies these characterizations to Einstein 4-orbifolds and develops an almost $ ext{RCD}$ framework to accommodate singular settings, offering local-to-global techniques via Poisson/heat-kernel estimates and cut-off construction. Finally, it outlines open questions on extending the structure theory to locally Ricci-bounded spaces, convergence questions, and quotients, highlighting potential future directions in the broader landscape of synthetic curvature-dimension theories.

Abstract

We provide various characterizations for a given almost smooth space to be an RCD space, in terms of a local volume doubling and a local Poincaré inequality. Applications include a characterization of Einstein $4$-orbifolds.

From almost smooth spaces to RCD spaces

TL;DR

The paper addresses how to locally characterize spaces on almost smooth metric measure spaces under a uniformly local PI framework, introducing the notions of SL and QL to connect local geometry with global curvature-dimension bounds. It proves two sharp equivalences: (i) an almost smooth space is iff it is PI with SL and QL and the -Bakry-Émery Ricci curvature bound holds on the smooth part, and (ii) if the singular set has codimension at least , the QL assumption can be dropped and the same characterization holds; these results extend to non-smooth frameworks. The work also applies these characterizations to Einstein 4-orbifolds and develops an almost framework to accommodate singular settings, offering local-to-global techniques via Poisson/heat-kernel estimates and cut-off construction. Finally, it outlines open questions on extending the structure theory to locally Ricci-bounded spaces, convergence questions, and quotients, highlighting potential future directions in the broader landscape of synthetic curvature-dimension theories.

Abstract

We provide various characterizations for a given almost smooth space to be an RCD space, in terms of a local volume doubling and a local Poincaré inequality. Applications include a characterization of Einstein -orbifolds.

Paper Structure

This paper contains 21 sections, 38 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Uniformly local condition; PI
  4. Main results
  5. Proof and organization
  6. Geometric analysis on metric measure space
  7. Metric measure space
  8. BE and RCD spaces
  9. Local objects
  10. PI condition
  11. Quantitative Lipschitz continuity of harmonic functions (QL)
  12. Local lower bound on Ricci curvature
  13. Locally BE spaces
  14. Locally RCD space
  15. Almost smooth metric measure space
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X$ be an $n$-dimensional almost smooth metric measure space (thus recall, in particular, it is complete), whose metric structure is a length space. Then, for all $K \in \mathbb{R}$ and $N \in [n, \infty)$, the following two conditions are equivalent.

Theorems & Definitions (107)

  • Theorem 1.1: Characterization of RCD for almost smooth space
  • Theorem 1.2: Characterization of RCD under codimension $4$-singularity
  • Definition 1: Metric measure space
  • Remark 1
  • Definition 2: $H^{1,2}$-Sobolev space
  • Definition 3: Minimal relaxed slope
  • Definition 4: Inifinitesimally Hilbertian (IH)
  • Remark 2
  • Definition 5: Laplacian
  • Definition 6: Heat flow
  • ...and 97 more