Scalar Curvature, Injectivity Radius and Immersions with Small Second Fundamental Forms

2022/03/26 by Misha Gromov, Gromov, Misha
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2203.14013

openalex publication_date 2022/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove in special cases the following. \bullet_Sc Bounds on the \it injectivity radii of "topologically complicated" Riemannian n-manifolds X, where the scalar curvatures of X are bounded from below, Sc(X)≥ σ>0. \bullet_curv Lower bounds on \it focal radii of smooth immersions from k-manifolds, e.g. homeomorphic to the k-torus, to certain Riemannian manifolds of dimensions n=k+m, e.g. to the cylinders S^n-1 × \mathbb R¹. \bullet_mean Topological lower bounds on the mean curvatures of domains in Riemannian manifolds. e.g. in the Euclidean n-space \mathbb Rⁿ. At the present moment, our results are limited by the \it spin condition and the \it n≤ 8 restriction.

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