On the positivity of a quasi-local mass in general dimensions

Abstract

In this paper, we obtain a positivity result of a quasi-local mass integral as proposed by Shi and Tam in general dimensions. The main argument is based on the monotonicity of a mass integral in a foliation of quasi-spherical metrics and a positive mass type theorem which was proved by Wang and Yau in the three-dimensional case, and is shown here in higher dimensions using spinor methods.The well-known positive mass theorem states that for a complete asymptotically flat manifold (M, g) such that g behaves like Euclidean at infinity near each end and suppose its scalar curvature is non-negative, then its ADM mass (cf.[2] for its definition and motivation) of each end is non-negative. Moreover, if the ADM mass of one of the end is zero, then (M, g) is actually the Euclidean space. The positive mass theorem was proved by Schoen and Yau [11,12] using minimal surface techniques. Later on, Witten [17] (see also [4,10]) gave a simplified proof the positive mass theorem using the spinors. Since then the method of spinors has been adopted by many people to prove positive mass-type theorems or some rigidity results; see for example [1,9,13,15,16].In particular, let us look at some results in this direction. Wang and Yau [15] developed a quasi-local mass for a three-dimensional manifold with boundary whose scalar curvature is bounded from below by some negative constant. Using spinors methods, they were able to prove that this mass is non-negative. Later on, Shi and Tam [14] also proved a similar result in the three-dimensional case, but with a simpler and more explicit definition of the mass. In this paper, we will show that the results of Shi-Tam and Wang-Yau also hold in higher dimensions.More

A note on the positivity of a quasi-local mass in general dimensions

“…Proof of Theorem 2.4. A similar positive mass type theorem when Σ can be isometrically embedded into H n −k 2 convexly was proved in [10] for n = 3, and was proved in [6] for general dimensions. The only difference in our case is that the metric g on the manifold M has two corners, near which the metric is only Lipschitz.…”

“…The first important ingredient is that for any future-directed null vector ζ ∈ R n,1 , there is a Killing spinor φ on (H n −k 2 , g hyp ) such that |φ| 2 g hyp = −2kX • ζ. It was proved in Propositions 2.1 and 2.2 of [6].…”

“…The only difference in our case is that the metric g on the manifold M has two corners, near which the metric is only Lipschitz. Since we are only interested in the asymptotic behavior, the argument in [10] and [6] can be carried through with minor modifications. We just describe the general idea of the proof in the following.…”

“…The first and third parts of the argument in higher dimensions are the same as in n = 3 except some obvious modification. For the positive mass type theorem in general dimensions, Kwong gave a detail proof of it in [6].…”

Boundary effect of Ricci curvature

“…Inequalities ( In what follows, by scaling the metric, we assume Ric ≥ (n − 1)g or Ric ≥ −(n − 1)g. In the latter case, the scalar curvature R of g satisfies R ≥ −n(n − 1). By the results in [20,18,9], there exists a sharp integral inequality relating H and | H H | if the manifold Ω is spin and the boundary Σ embeds isometrically in the hyperbolic space H n as a convex hypersurface. On the other hand, the counterexample to Min-Oo's conjecture in [1] shows that even the pointwise condition H = | H S | is not sufficient to guarantee rigidity if one only assumes R ≥ n(n − 1).…”