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Cohomogeneity one RCD-spaces

Diego Corro, Jesús Núñez-Zimbrón, Jaime Santos-Rodríguez

TL;DR

This paper develops a comprehensive framework for cohomogeneity-one actions on $ mathsf{RCD}$ spaces, establishing a Slice Theorem in the non-collapsed setting and analyzing induced infinitesimal actions on tangent cones to obtain a precise cone-structure for slices. It then builds a constructive correspondence between cohomogeneity-one group diagrams and $\mathsf{RCD}$ spaces, proving a complete description of cohomogeneity-one $\mathsf{RCD}$ spaces and a topological rigidity theory that yields explicit models in all quotient-geometry cases. Using warped-product and gluing techniques, the authors develop methods to generate new cohomogeneity-one $\mathsf{RCD}$ spaces from group data, including curvature-dimension preservation via $\mathcal{F}K$-concavity and $CD^*$ bounds. They apply these tools to classify non-collapsed cohomogeneity-one $\mathsf{RCD}$ spaces of essential dimension at most $4$, showing they are Alexandrov spaces in these cases, and they present refined constructions and higher-dimensional examples that demonstrate the sharpness of the low-dimension results. The work thus extends Grove–Ziller symmetry principles to synthetic lower Ricci bounds, providing a complete, constructive, and dimension-sensitive theory for cohomogeneity-one $\mathsf{RCD}$ spaces.

Abstract

We study $\mathsf{RCD}$-spaces $(X,d,\mathfrak{m})$ with group actions by isometries preserving the reference measure $\mathfrak{m}$ and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that when $X$ is non-collapsed the slices are homeomorphic to metric cones over homogeneous spaces with $\mathrm{Ric} \geq 0$. As a consequence we obtain complete topological structural results (also in the collapsed case) and a regular orbit representation theorem. Conversely, we show how to construct new $\mathsf{RCD}$-spaces from a cohomogeneity one group diagram, giving a complete description of $\mathsf{RCD}$-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed $\mathsf{RCD}$-spaces of essential dimension at most $4$.

Cohomogeneity one RCD-spaces

TL;DR

This paper develops a comprehensive framework for cohomogeneity-one actions on spaces, establishing a Slice Theorem in the non-collapsed setting and analyzing induced infinitesimal actions on tangent cones to obtain a precise cone-structure for slices. It then builds a constructive correspondence between cohomogeneity-one group diagrams and spaces, proving a complete description of cohomogeneity-one spaces and a topological rigidity theory that yields explicit models in all quotient-geometry cases. Using warped-product and gluing techniques, the authors develop methods to generate new cohomogeneity-one spaces from group data, including curvature-dimension preservation via -concavity and bounds. They apply these tools to classify non-collapsed cohomogeneity-one spaces of essential dimension at most , showing they are Alexandrov spaces in these cases, and they present refined constructions and higher-dimensional examples that demonstrate the sharpness of the low-dimension results. The work thus extends Grove–Ziller symmetry principles to synthetic lower Ricci bounds, providing a complete, constructive, and dimension-sensitive theory for cohomogeneity-one spaces.

Abstract

We study -spaces with group actions by isometries preserving the reference measure and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that when is non-collapsed the slices are homeomorphic to metric cones over homogeneous spaces with . As a consequence we obtain complete topological structural results (also in the collapsed case) and a regular orbit representation theorem. Conversely, we show how to construct new -spaces from a cohomogeneity one group diagram, giving a complete description of -spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed -spaces of essential dimension at most .
Paper Structure (20 sections, 68 theorems, 149 equations, 1 table)

This paper contains 20 sections, 68 theorems, 149 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calculus on metric measure spaces
  4. Bakry-Émery condition
  5. Non-collapsed spaces
  6. Warped products
  7. Transformation groups
  8. Equivariant Gromov-Hausdorff convergence
  9. Slice Theorem for cohomogeneity one geodesic spaces
  10. Cohomogeneity one Group diagrams
  11. Geometry of the slice
  12. Induced infinitesimal actions
  13. Topological Rigidity
  14. Construction of cohomogeneity one RCD-spaces
  15. Cone-like warped products.
  16. ...and 5 more sections

Key Result

Theorem 1

þ Let $(X,d,\mathfrak{m})$ be an $\mathsf{RCD}(K,N)$-space. Let $G$ be a compact Lie group acting on $X$ by measure preserving isometries and cohomogeneity one. Then the following hold: Moreover the cone fibers in items $(a)$ and $(b)$ are cones over homogeneous spaces. In the case when $X$ is non-collapsed then the cone fibers admit metric cone structures over $\mathsf{RCD}(N-k_{\pm}-2,N-k_\pm-1

Theorems & Definitions (149)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 2.1: Cheeger energy
  • Definition 2.2: Sobolev space
  • Remark 2.3
  • Definition 2.4: Pointwise inner product
  • ...and 139 more