On the Fill-in of Nonnegative Scalar Curvature Metrics

Abstract

In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $(\Sigma ,\gamma ,H)$. We prove that given a metric $\gamma$ on ${\mathbf{S}}^{n-1}$ ($3\le n\le 7$), $({\mathbf{S}}^{n-1},\gamma ,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough (Theorem \ref{Thm: no fillin nonnegative scalar 2}). Moreover, we prove that if $\gamma$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on ${\mathbf{S}}^{n-1}$, then the much weaker condition that the total mean curvature ${\int }_{{\mathbf{S}}^{n-1}}H\mathrm{d}{\mu }_{\gamma }$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P.\,23 in \cite{Gromov4}). In the second part of this paper, we investigate the $\theta$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

34 pages, all comments are welcome

Cited by in corpus (2)

• NNSC-Cobordism of Bartnik Data in High Dimensions
• Nonexistence of NNSC fill-ins with large mean curvature