On the Fill-in of Nonnegative Scalar Curvature Metrics

2019/07/29 by Shi, Yuguang, Wang, Wenlong, Wei, Guodong +1
#Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C20 #Secondary 83C99

paper · doi:10.48550/arxiv.1907.12173

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 (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 \refThm: no fillin nonnegative scalar 2). 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 ∫_{\mathbf S^n-1}H \mathrm dμ_γ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P. 23 in \citeGromov4). 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.

Related

• Noncompact Fill-Ins of Bartnik Data
• Nonexistence of NNSC-cobordism of Bartnik data
• Nonexistence of NNSC fill-ins with large mean curvature
• Total mean curvature of the boundary and nonnegative scalar curvature fill-ins
• Fill-Ins of Tori with Scalar Curvature Bounded from Below