Positive mass theorems of ALF and ALG manifolds

2021/03/21 by Peng Liu, Liu, Peng, Yuguang Shi +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary 53C20 #Secondary 83C99 #math.DG #msc:53C20 #msc:83C99

paper · pdf · doi:10.48550/arxiv.2103.11289

32pages, 5 figures, all comments are welcome!

openalex publication_date 2021/03/21 · arxiv created 2021/03/23 · arxiv updated 2021/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we want to prove positive mass theorems for ALF and ALG manifolds with model spaces \mathbb R^n-1× \mathbb S¹ and \mathbb R^n-2× \mathbb T² respectively in dimensions no greater than 7 (Theorem \refALFPMT0). Different from the compatibility condition for spin structure in \cite[Theorem 2]minerbe2008a, we show that some type of incompressible condition for \mathbb S¹ and \mathbb T² is enough to guarantee the nonnegativity of the mass. As in the asymptotically flat case, we reduce the desired positive mass theorems to those ones concerning non-existence of positive scalar curvature metrics on closed manifolds coming from generalize surgery to n-torus. Finally, we investigate certain fill-in problems and obtain an optimal bound for total mean curvature of admissible fill-ins for flat product 2-torus \mathbb S¹(l₁)× \mathbb S¹(l₂).

Citations

• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

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• Mass Lower Bounds for Asymptotically Locally Flat Manifolds

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