| _version_ | 1866901856596262912 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper explores the burgeoning field of singular Riemannian manifolds endowed with lower curvature bounds. While classical Riemannian geometry primarily focuses on smooth manifolds, many natural geometric limits and constructions, such as collapsing sequences, metric cones, or spaces arising from non-linear PDEs, inherently exhibit singularities. The presence of these singularities poses significant challenges to applying traditional differential geometric tools. However, the pioneering work in metric geometry and synthetic notions of curvature has enabled the development of a robust structure theory for such spaces. This work delves into the fundamental concepts and methodologies, including Alexandrov spaces with lower sectional curvature bounds and metric measure spaces with lower Ricci curvature bounds (RCD spaces). We discuss the analytical and topological tools, such as Gromov-Hausdorff convergence, optimal transport, and heat kernel analysis, that are crucial for understanding the intrinsic geometry and local behavior of these singular spaces. Key results, including stratification theorems, regularity properties, and stability under geometric limits, are examined. The paper aims to synthesize the current understanding, highlight major achievements, and outline promising avenues for future research in this dynamic area of geometric analysis. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17711376 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Structure Theory for Singular Riemannian Manifolds with Lower Curvature Bounds Revista, Zen MATH, 10 This paper explores the burgeoning field of singular Riemannian manifolds endowed with lower curvature bounds. While classical Riemannian geometry primarily focuses on smooth manifolds, many natural geometric limits and constructions, such as collapsing sequences, metric cones, or spaces arising from non-linear PDEs, inherently exhibit singularities. The presence of these singularities poses significant challenges to applying traditional differential geometric tools. However, the pioneering work in metric geometry and synthetic notions of curvature has enabled the development of a robust structure theory for such spaces. This work delves into the fundamental concepts and methodologies, including Alexandrov spaces with lower sectional curvature bounds and metric measure spaces with lower Ricci curvature bounds (RCD spaces). We discuss the analytical and topological tools, such as Gromov-Hausdorff convergence, optimal transport, and heat kernel analysis, that are crucial for understanding the intrinsic geometry and local behavior of these singular spaces. Key results, including stratification theorems, regularity properties, and stability under geometric limits, are examined. The paper aims to synthesize the current understanding, highlight major achievements, and outline promising avenues for future research in this dynamic area of geometric analysis. |
| title | Structure Theory for Singular Riemannian Manifolds with Lower Curvature Bounds |
| url | https://doi.org/10.5281/zenodo.17711376 |