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 $$\left( {\sum _1^{n - 1},{{\rm{\gamma }}_1},{H_1}} \right)$$ and $$\left( {\sum _2^{n - 1},{{\rm{\gamma }}_2},{H_2}} \right)$$ . 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 $$\int_{{S^2}}^{} {{H_2}d{\mu _{{\gamma _2}}}} $$ 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 $$\left( {\Sigma _1^{n - 1},{{\rm{\gamma }}_1},0} \right)$$ and $$\left( {{\rm{\Sigma }}_2^{n - 1},{{\rm{\gamma }}_2},{H_2}} \right)$$ admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough (see Theorem 1.5).