Nonexistence of NNSC-cobordism of Bartnik data

TL;DR

This paper proves nonexistence results for nonnegative scalar curvature cobordisms between Bartnik data under certain large mean curvature conditions, extending to higher dimensions and specific topologies.

Contribution

It establishes new nonexistence theorems for NNSC cobordisms with large prescribed mean curvature, including for higher dimensions and specific geometric conditions.

Findings

No NNSC cobordism exists for large mean curvature in higher dimensions.

Large total mean curvature rules out NNSC cobordisms in 2D case.

Positivity of Gaussian curvature and mass criteria prevent trivial NNSC cobordisms.

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 $\int_{S^2}H_2d\mu_{\gamma_2}$ 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}).