2020/03/03 by Jintian Zhu, Zhu, Jintian · 7 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Primary 53C21 #Secondary 53C24
paper · pdf · doi:10.48550/arxiv.2003.01315
openalex publication_date 2020/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open 3-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be \mathbf R2× (-c,c) with some doubly warped product metric if the optimal bound is attained. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses in 3-spheres with positive sectional curvatures.