Positive mass theorems of ALF and ALG manifolds

2021/03/21 by Liu, Peng, Shi, Yuguang, Zhu, Jintian · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C20 #Secondary 83C99

paper · doi:10.48550/arxiv.2103.11289

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).

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