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 $$(\varSigma ,\gamma ,H)$$ . 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 4). 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 (Four lectures on scalar curvature, 2019, see P. 23). 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.