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On the Structure of Busemann Spaces with Non-Negative Curvature

Bang-Xian Han, Liming Yin

TL;DR

This work advances the synthetic geometry of non-smooth spaces by developing a fine structure theory for finite-dimensional Busemann spaces with non-negative curvature under $S$-concavity and local semi-convexity. It demonstrates the existence of a non-trivial integer-dimensional Hausdorff measure $\mathcal{H}^n$, the measure contraction property $\mathrm{MCP}(0,n)$, and rectifiability, with almost every point possessing a unique tangent cone isometric to a finite-dimensional Banach space; under mild uniform convexity/smoothness, Banach tangent cones exhibit 2-uniform convexity and uniform smoothness. The authors introduce two notions of angle, two flavors of strainer maps, and a novel dequeuing strategy to achieve self-improvement of strainers, enabling precise control of regular and singular sets. Consequently, they derive a stratification with sharp Hausdorff-dimension bounds for singular strata and establish a robust link to Finslerian metric-measure spaces through Banach-tangent-cones, enriching the synthetic-curvature toolkit beyond Alexandrov spaces. The results provide tools and examples for studying metric-measure spaces with synthetic sectional curvature lower bounds, with potential applications to Finsler geometry and related non-Riemannian contexts.

Abstract

We extend the structure theory of Burago--Gromov--Perelman for Alexandrov spaces with curvature bounded below, to the setting of Busemann spaces with non-negative curvature. We prove that any finite-dimensional Busemann space with non-negative curvature satisfying Ohta's $S$-concavity and local semi-convexity, admits a non-trivial integer-dimensional Hausdorff measure, and satisfies the measure contraction property. We also show that such spaces are rectifiable and that almost every point admits a unique tangent cone isometric to a finite-dimensional Banach space. In addition, under mild control of the uniform smoothness constant, we obtain refined estimates for the Hausdorff dimension of the singular strata. Our results not only enrich the theory of synthetic sectional curvature lower bound for metric spaces, but also provide some useful tools and examples to study Finslerian metric measure spaces.

On the Structure of Busemann Spaces with Non-Negative Curvature

TL;DR

This work advances the synthetic geometry of non-smooth spaces by developing a fine structure theory for finite-dimensional Busemann spaces with non-negative curvature under -concavity and local semi-convexity. It demonstrates the existence of a non-trivial integer-dimensional Hausdorff measure , the measure contraction property , and rectifiability, with almost every point possessing a unique tangent cone isometric to a finite-dimensional Banach space; under mild uniform convexity/smoothness, Banach tangent cones exhibit 2-uniform convexity and uniform smoothness. The authors introduce two notions of angle, two flavors of strainer maps, and a novel dequeuing strategy to achieve self-improvement of strainers, enabling precise control of regular and singular sets. Consequently, they derive a stratification with sharp Hausdorff-dimension bounds for singular strata and establish a robust link to Finslerian metric-measure spaces through Banach-tangent-cones, enriching the synthetic-curvature toolkit beyond Alexandrov spaces. The results provide tools and examples for studying metric-measure spaces with synthetic sectional curvature lower bounds, with potential applications to Finsler geometry and related non-Riemannian contexts.

Abstract

We extend the structure theory of Burago--Gromov--Perelman for Alexandrov spaces with curvature bounded below, to the setting of Busemann spaces with non-negative curvature. We prove that any finite-dimensional Busemann space with non-negative curvature satisfying Ohta's -concavity and local semi-convexity, admits a non-trivial integer-dimensional Hausdorff measure, and satisfies the measure contraction property. We also show that such spaces are rectifiable and that almost every point admits a unique tangent cone isometric to a finite-dimensional Banach space. In addition, under mild control of the uniform smoothness constant, we obtain refined estimates for the Hausdorff dimension of the singular strata. Our results not only enrich the theory of synthetic sectional curvature lower bound for metric spaces, but also provide some useful tools and examples to study Finslerian metric measure spaces.

Paper Structure

This paper contains 24 sections, 40 theorems, 89 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation and object
  3. Main results
  4. Main ideas and tools
  5. Organization of the paper
  6. Preliminaries and notations
  7. Spaces and maps
  8. Hausdorff measure, dimensions and rectifiability
  9. Gromov--Hausdorff convergence and blow-ups
  10. Uniform convexity and smoothness
  11. S-concavity, local semi-convexity and Busemann concavity
  12. Angles
  13. Angles viewed from a fixed point
  14. Angles of fixed scale and tangent cones
  15. Strainers and strainer maps
  16. ...and 9 more sections

Key Result

Theorem 1.5

Let $(X,d)$ be an $S$-concave, locally semi-convex, Busemann concave space for some $S \geq 1$. Then $X$ is of finite Hausdorff dimension if and only if it has finite strainer number. In either case, both values coincide with the topological dimension of $X$. Moreover, $X$ admits a non-trivial Hausd

Theorems & Definitions (103)

  • Definition 1.1: Busemann convex
  • Definition 1.2: Busemann convex
  • Definition 1.3: Busemann concave
  • Theorem 1.5: Proposition \ref{['prop:Buseman_concave_MCP']} and Corollary \ref{['cor:strained_points_open_dense_top_manifold']}
  • Theorem 1.6: Theorem \ref{['thm:full_measure_n_strained_points']}
  • Theorem 1.7: Theorem \ref{['thm:n_rect']} and \ref{['thm:Banach_tangent_cone']}, Corollary \ref{['cor:characterize_Banach_tangent_cone']}
  • Theorem 1.8: Theorem \ref{['thm:main_thm_Hausdorff_dim_est']}, Corollary \ref{['cor:stratification']}
  • Remark 1.9
  • Definition 3.1: $S$-concave
  • Remark 3.2
  • ...and 93 more