Nonexistence of NNSC-cobordism of Bartnik data

Abstract

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 ${\int }_{S^2}{H}_{2}d{\mu }_{{\gamma }_2}$ 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}).

17pages, All comments are welcome! The paper has been accepted for publication in SCIENCE CHINA Mathematics

References in corpus (3)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
• NNSC-Cobordism of Bartnik Data in High Dimensions
• Nonexistence of NNSC fill-ins with large mean curvature