Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

Abstract

In the first part of the paper, we get some estimates for the supremum of the total mean curvature of boundaries of domains with nonnegative scalar curvature, and discuss its relationship with the positive mass theorem of asymptotically flat (hyperbolic) manifolds. In the second part of the paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with nonempty boundaries which completely solve an open problem due to Gromov (see Question 1.1). The results in this paper also provide some partially affirmative answers to Gromov's conjectures formulated in [15] (see Conjecture 1.1, Conjecture 1.2 below).

On a spacetime positive mass theorem with corners

“…By [57] Lemma 2.1, we know that there exists a cobordism (Σ × [0, 1], ĝ) and h 0 , h 1 ∈ C ∞ (Σ) such that (1) ĝ| Σ×{0} = γ and ĝ| Σ×{1} = γ 1 , (2) With respect to ĝ and the outward normal, the mean curvature of Σ × {0} and Σ × {1} are respectively h 0 and h 1 , (3) h 1 > h and (4) R ĝ > 0. Pick C 0 = max(−h 0 ).…”

“…The regularity assumptions of (g, k) in Theorem 1.1 naturally arise from the fill-in and extension problems (e.g. [8], [44], [54], [58], [57]). For example, let (M 1 , g 1 ), (M 2 , g 2 ) be two Riemannian manifolds with smooth boundary, where ∂M 1 is isometric to ∂M 2 .…”

“…While in [57], the parabolic method to extend metric in [54] is used to construct a PSC (R g > 0) collar, which combined with he Riemannian positive mass theorem with corners can show non-existence of NNSC fill-ins. In the same spirit as above, we can arrive at the following conclusion.…”

“…In [58] and [57], there was a partial affirmative answer given by the parabolic method to construct a metric extension done in [54]. It is shown that if the mean curvature is too large and an NNSC (non-negative scalar curvature) fillin to a Bartnik data set exists, then there is a contradiction to the Riemannian positive mass theorem with corners.…”

“…(cf [57]. Theorem 1.2) Let D SB := (Σ n−1 , γ, α, H, β) be a spacetime Bartnik data set where Σ n−1 can be smoothly embedded into R n and γ is smooth.…”