NNSC-Cobordism of Bartnik Data in High Dimensions

Abstract

In this short note, we formulate three problems relating to nonnegative scalar curvature (NNSC) fill-ins. Loosely speaking, the first two problems focus on: When are (n−1)-dimensional Bartnik data (Σⁿ⁻¹ᵢ, γᵢ, Hᵢ), i=1,2, NNSC-cobordant? If (𝕊ⁿ⁻¹, γₛₜd, 0) is positive scalar curvature (PSC) cobordant to (Σⁿ⁻¹₁,γ₁, H₁), where (𝕊ⁿ⁻¹, γₛₜd) denotes the standard round unit sphere, then (Σⁿ⁻¹₁,γ₁, H₁) admits an NNSC fill-in. Just as Gromov's conjecture is connected with the positive mass theorem, our problems are connected with the Penrose inequality, at least in the case of n=3. Our third problem is on Λ(Σⁿ⁻¹, γ) defined below.