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Register a new userOn the Fill-in of Nonnegative Scalar Curvature Metrics
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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 $mathbf{S}^{n-1}$ ($3leq nleq 7$), $(mathbf{S}^{n-1},gamma,H)$ 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 $int_{mathbf S^{n-1}}H,mathrm dmu_gamma$ 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.
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In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, which completely solves an open problem due to Gromov (see Question ref{extension1}). Then we introduce a fill-in invariant (see Definition ref{fillininvariant}) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromovs conjectures formulated in cite{Gro19} (see Conjecture ref{conj0} and Conjecture ref{conj1} below)
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