On the fill-in of nonnegative scalar curvature metrics

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 ( Σ , γ , H ) . We prove that given a metric γ on S n - 1 ( 3 ≤ n ≤ 7 ), ( S n - 1 , γ , H ) admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if γ is a positive scalar curvature (PSC) metric isotopic to the standard metric on S n - 1 , then the much weaker condition that the total mean curvature ∫ S n - 1 H d μ γ 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 θ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.