Busemann and MCP

TL;DR

The paper investigates Busemann spaces with MCP and proves rigidity results: geodesically complete MCP($0,N$) spaces are strictly convex Banach spaces of dimension $n\le N$, with equality under non-collapsing; in the collapsed setting, the space embeds as a closed convex subset of such a Banach space. It also establishes a manifold-with-boundary structure under local non-collapsed MCP($K,n$) with $K\le0$, showing the interior consists of $n$-regular points and tangent cones are unique; the interior inherits almost-Riemannian coordinates via an almost-isometric exponential map. A key strategy combines Andreev’s cone-type rigidity, measure-homogeneity arguments, tangent-cone continuity, and non-collapsed MCP/MRR25 results to derive both global and local rigidity and almost rigidity results for the interior, and to analyze the boundary via a Toponogov-type globalization. The work situates Busemann+MCP spaces within the broader CAT/CD framework, extends rigidity phenomena to Finsler-type settings, and highlights open problems about collapsing cases, Finsler structures on interiors, and Berwald-type properties with MCP/CD. These findings advance the understanding of synthetic curvature bounds beyond CAT(RCD) spaces and provide a framework for almost-Riemannian structure in non-smooth settings.

Abstract

We study the structure of Busemann spaces with measures satisfying the measure contraction property (MCP). The main results are rigidity theorems and structure theorems under the assumption of geodesic completeness or non-collapse. The appendix contains some observations on the tangent cones of geodesically complete Busemann spaces.

Figures (8)

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Theorems & Definitions (102)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Definition 2.1
• Remark 2.2
• Example 2.3