NNSC-Cobordism of Bartnik Data in High Dimensions

Abstract

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 $({\Sigma }_i^{n-1},{\gamma }_i,H_i)$, $i=1,2$, NNSC-cobordant? (i.e., there is an $n$-dimensional compact Riemannian manifold $({\Omega }^n,g)$ with scalar curvature $R(g)\ge 0$ 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 $({\Omega }^n,g)$). If $({\mathbb{S}}^{n-1},{\gamma }_{\mathrm{std}},0)$ is positive scalar curvature (PSC) cobordant to $({\Sigma }_1^{n-1},{\gamma }_1,H_1)$, where $({\mathbb{S}}^{n-1},{\gamma }_{\mathrm{std}})$ denotes the standard round unit sphere then $({\Sigma }_1^{n-1},{\gamma }_1,H_1)$ 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 $\Lambda ({\Sigma }^{n-1},\gamma )$ defined below.

References in corpus (2)

• Metrics of positive scalar curvature and generalised Morse functions, part 1
• On the Fill-in of Nonnegative Scalar Curvature Metrics

Cited by in corpus (1)

• Nonexistence of NNSC-cobordism of Bartnik data