Positive mass theorems of ALF and ALG manifolds

In this paper, we want to prove positive mass theorems for ALF and ALG manifolds with model spaces ${\mathbb{R}}^{n-1}\times {\mathbb{S}}^1$ and $\mathbb R^{n-2}\times \mathbb T^2$respectivelyindimensionsnogreaterthan$7$ (Theorem \ref{ALFPMT0}). { Different from the compatibility condition for spin structure in \cite[Theorem 2]{minerbe2008a}, we show that some type of incompressible condition for ${\mathbb{S}}^1$ and ${\mathbb{T}}^2$ 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 ${\mathbb{S}}^1(l_1)\times {\mathbb{S}}^1(l_2)$.}