A topological gap theorem for the π₂-systole of positive scalar curvature 3-manifolds

2023/07/04 by Kai Xu, Xu, Kai · 2 citations
Mathematics · Computer Science · #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2307.01922

Abstract

Let M be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If M has nonvanishing second homotopy group, then it is known that the π₂-systole of M (i.e. the minimal achievable area of homotopically nontrivial spheres) is at most 8π. We prove the following gap theorem: if M is further not a quotient of S²× S¹, then the π₂-systole of M is no greater than an improved constant c≈ 5.44π. This statement follows as a new topological application of Huisken and Ilmanen's weak inverse mean curvature flow.

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• A 2-systolic inequality on non-rational compact Kähler surfaces with positive scalar curvature
• Minimal surfaces and comparison geometry

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