Nonexistence of NNSC-cobordism of Bartnik data

In this paper, we consider the problem of nonnegative scalar curvature (NNSC) cobordism of Bartnik data $(Σ_1^{n-1}, γ_1, H_1)$ and $(Σ_2^{n-1}, γ_2, H_2)$. We prove that given two metrics $γ_1$ and $γ_2$ on $S^{n-1}$ ($3\le n\le 7$) 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(Theorem \ref{highdimnoncob0}). Moreover, we show that for $n=3$, a much weaker condition that the total mean curvature $\int_{S^2}H_2dμ_{γ_2}$ is large enough rules out NNSC cobordisms(Theorem \ref{2-d0}); if we require the Gaussian curvature of $γ_2$ 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 $(Σ_1^{n-1}, γ_1, 0)$ and $(Σ_2^{n-1}, γ_2, H_2)$ admit no NNSC cobordism provided the prescribed mean curvature $H_2$ is large enough(Theorem \ref{highdimnoncob10}).

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