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 γ₁ and γ₂ on Sⁿ⁻¹ (3 ⩽ n ⩽ 7) with H₁ fixed, then (Sⁿ⁻¹, γ₁, H₁) and (Sⁿ⁻¹, γ₂, H₂) admit no NNSC-cobordism provided the prescribed mean curvature H₂ 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 γ₂ 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 H₂ is large enough (see Theorem 1.5). [ABSTRACT FROM AUTHOR]