Almost-Riemannian manifolds do not satisfy the curvature-dimension condition

Journal article

Abstract The Lott–Sturm–Villani curvature-dimension condition $$\textsf{CD}(K,N)$$ CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N . It was proved by Juillet (Rev Mat Iberoam 37(1), 177–188, 2021) that a large class of sub-Riemannian manifolds do not satisfy the $$\textsf{CD}(K,N)$$ CD(K,N) condition, for any $$K\in {\mathbb {R}}$$ K∈R and $$N\in (1,\infty )$$ N∈(1,∞) . However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the $$\textsf{CD}$$ CD condition in this setting, providing a new strategy which allows us to contradict the one-dimensional version of the $$\textsf{CD}$$ CD condition. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the $$\textsf{CD}(K,N)$$ CD(K,N) condition for any $$K\in {\mathbb {R}}$$ K∈R and $$N\in (1,\infty )$$ N∈(1,∞) .

Authors: Magnabosco, M; Rossi, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Calculus of Variations and Partial Differential Equations
• Volume: 62
• Issue: 4
• Pages: 123-123
• Article number: 123
• Publication date: 2023-03-20
• DOI: 10.1007/s00526-023-02466-x
• EISSN: 1432-0835
• ISSN: 0944-2669
• Language: English
• Keywords: Minimal volume; Ricci-flat manifold; Sectional curvature; Mathematics; Pure mathematics; Scalar curvature; Riemannian manifold; Geometry; Mathematical analysis; Prescribed scalar curvature problem; Dimension (graph theory); Curvature; Curvature of Riemannian manifolds