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 R × S and R × T respectively in dimensions no greater than 7 (Theorem 1.2). Different from the compatibility condition for spin structure in [20, Theorem 2], we show that some type of incompressible condition for S and T 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 nonexistence of positive scalar curvature metrics on closed manifolds coming from generalize surgery to ntorus. 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 S(l1)× S (l2).

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