Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 030, 5 pages      arXiv:2001.05607
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

NNSC-Cobordism of Bartnik Data in High Dimensions

Xue Hu a and Yuguang Shi b
a) Department of Mathematics, College of Information Science and Technology, Jinan University, Guangzhou, 510632, P.R. China
b) Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China

Received January 22, 2020, in final form April 13, 2020; Published online April 20, 2020

In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are $(n-1)$-dimensional Bartnik data $\big(\Sigma_i ^{n-1}, \gamma_i, H_i\big)$, $i=1,2$, NNSC-cobordant? (i.e., there is an $n$-dimensional compact Riemannian manifold $\big(\Omega^n, g\big)$ with scalar curvature $R(g)\geq 0$ and the boundary $\partial \Omega=\Sigma_{1} \cup \Sigma_{2}$ such that $\gamma_i$ is the metric on $\Sigma_i ^{n-1}$ induced by $g$, and $H_i$ is the mean curvature of $\Sigma_i$ in $\big(\Omega^n, g\big)$). If $\big(\mathbb{S}^{n-1},\gamma_{\rm std},0\big)$ is positive scalar curvature (PSC) cobordant to $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$, where $\big(\mathbb{S}^{n-1}, \gamma_{\rm std}\big)$ denotes the standard round unit sphere then $\big(\Sigma_1 ^{n-1}, \gamma_1, H_1\big)$ admits an NNSC fill-in. Just as Gromov's conjecture is connected with positive mass theorem, our problems are connected with Penrose inequality, at least in the case of $n=3$. Our third problem is on $\Lambda\big(\Sigma^{n-1}, \gamma\big)$ defined below.

Key words: scalar curvature; NNSC-cobordism; quasi-local mass; fill-ins.

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