In this paper, we consider the problem of nonnegative scalar curvature (NNSC) cobordism of Bartnik data $(\Sigma_1^{ n-1}, \gamma_1, H_1)$ and $(\Sigma_2^{ n-1}, \gamma_2, H_2)$. We prove that given two metrics $\gamma_1$ and $\gamma_2$ on $S^{ n-1}$ ($3\le n\le 7$) with $H_1$ fixed, then $(S^{ n-1}, \gamma_1, H_1)$ and $(S^{ n-1}, \gamma_2, H_2)$ admit no NNSC cobordism provided the prescribed mean curvature $H_2$ is large enough(Theorem \ref{ highdimnoncob0}). Moreover, we show that for $n=3$, a much weaker condition that the total mean curvature $\int_{ S^2}H_2d\mu_{ \gamma_2}$ is large enough rules out NNSC cobordisms(Theorem \ref{ 2-d0}); if we require the Gaussian curvature of $\gamma_2$ to be positive, we get a criterion for non existence of trivial NNSC-cobordism by using Hawking mass and Brown-York mass(Theorem \ref{ cobordism20}). For the general topology case, we prove that $(\Sigma_1^{ n-1}, \gamma_1, 0)$ and $(\Sigma_2^{ n-1}, \gamma_2, H_2)$ admit no NNSC cobordism provided the prescribed mean curvature $H_2$ is large enough(Theorem \ref{ highdimnoncob10}).
Categories: math.DG