2003/02/20 by Christopher B. Croke, Croke, Christopher B., Mikhail G. Katz +1
Mathematics · #53C23 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #History and Overview (math.HO) #Point processes and geometric inequalities #math.DG #math.GT #math.HO #msc:53C23
paper · pdf · doi:10.48550/arxiv.math/0302248
34 pages. To appear in Surveys in Differential Geometry
arxiv created 2003/02/20 · openalex publication_date 2003/02/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By `universal' we mean without curvature assumptions. The restriction to results with no (or only minimal) curvature assumptions, although somewhat arbitrary, allows the survey to be reasonably short. Although, even in this limited case the authors have left out many interesting results.