Nonexistence of NNSC-cobordism of Bartnik data

2020/10/31 by Bo, Leyang, Shi, Yuguang
#53C20 #83C99 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2011.00204

Abstract

In this paper, we consider the problem of nonnegative scalar curvature (NNSC) cobordism of Bartnik data (Σ₁^n-1, γ₁, H₁) and (Σ₂^n-1, γ₂, H₂). We prove that given two metrics γ₁ and γ₂ on S^n-1 (3≤ n≤ 7) with H₁ fixed, then (S^n-1, γ₁, H₁) and (S^n-1, γ₂, H₂) admit no NNSC cobordism provided the prescribed mean curvature H₂ is large enough(Theorem \refhighdimnoncob0). Moreover, we show that for n=3, a much weaker condition that the total mean curvature ∫_S²H₂dμ_γ2 is large enough rules out NNSC cobordisms(Theorem \ref2-d0); if we require the Gaussian curvature of γ₂ to be positive, we get a criterion for non existence of trivial NNSC-cobordism by using Hawking mass and Brown-York mass(Theorem \refcobordism20). For the general topology case, we prove that (Σ₁^n-1, γ₁, 0) and (Σ₂^n-1, γ₂, H₂) admit no NNSC cobordism provided the prescribed mean curvature H₂ is large enough(Theorem \refhighdimnoncob10).

Related

• On the Fill-in of Nonnegative Scalar Curvature Metrics
• Noncompact Fill-Ins of Bartnik Data
• Nonexistence of NNSC fill-ins with large mean curvature
• Nonnegative curvature, elliptic genus and unbounded Pontryagin numbers
• Cobordism Invariance of the Homotopy Type of the Space of Positive\n Scalar Curvature Metrics