Nonexistence of the NNSC-cobordism of Bartnik data

In this paper, we consider the problem of the nonnegative scalar curvature (NNSC)-cobordism of Bartnik data ( ∑ 1 n − 1 , γ 1 , H 1 ) and ( ∑ 2 n − 1 , γ 2 , H 2 ) . We prove that given two metrics γ 1 and γ 2 on S n −1 (3 ⩽ n ⩽ 7) with H 1 fixed, then ( S n −1 , γ 1 , H 1 ) and ( S n −1 , γ 2 , H 2 ) admit no NNSC-cobordism provided the prescribed mean curvature H 2 is large enough (see Theorem 1.3). Moreover, we show that for n = 3, a much weaker condition that the total mean curvature ∫ S 2 H 2 d μ γ 2 is large enough rules out NNSC-cobordisms (see Theorem 1.2); if we require the Gaussian curvature of γ 2 to be positive, we get a criterion for nonexistence of the trivial NNSC-cobordism by using the Hawking mass and the Brown-York mass (see Theorem 1.1). For the general topology case, we prove that ( Σ 1 n − 1 , γ 1 , 0 ) and ( Σ 2 n − 1 , γ 2 , H 2 ) admit no NNSC-cobordism provided the prescribed mean curvature H 2 is large enough (see Theorem 1.5).