Measuring Mass via Coordinate Cubes
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Communications in
Mathematical Physics
Measuring Mass via Coordinate Cubes Pengzi Miao Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA. E-mail: [email protected] Received: 8 January 2020 / Accepted: 28 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat 3-manifold along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov’s scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning angle of slicing curves, the formula realizes the mass as integration of the angle defect detected by the boundary term in the Gauss–Bonnet theorem.
1. Motivation and Mass Formulae In [12], Stern gave an intriguing formula relating the scalar curvature of a manifold to the level set of its harmonic functions. In its simplest form, Stern’s formula [12, equation (14)] shows |∇u| =
1 2 2 |∇ u| + |∇u|2 (R − 2K ) 2|∇u|
(1)
near points where ∇u = 0, here u is a harmonic function on a Riemannian 3-manifold (M 3 , g), R and K denote the scalar curvature of g and the Gauss curvature of , the level set of u, respectively. Applications of the formula to closed manifolds and to compact manifolds with boundary were given by Stern [12], and Bray and Stern [4]. If the manifold (M 3 , g) is asymptotically flat, by applying Stern’s formula, Bray et al. [3] gave a new elegant proof of the 3-dimensional positive mass theorem, which was originally proved by Schoen and Yau [11], and Witten [13]. Moreover, the result in [3] provides an explicit lower bound of the mass of (M, g) via a single harmonic function. Pengzi Miao research was partially supported by NSF Grant DMS-1906423.
P. Miao
In the context of asymptotically flat manifolds, an observation of Bartnik [2] was 3 i=1
S∞
1 ∂ |∇ y i |2 dσ = 16π m(g), 2 ∂ν
(2)
where m(g) is the mass of (M, g), {y i } are harmonic coordinates near infinity, and S∞ denotes the limit of integration along a sequence of suitable surfaces tending to infinity. As |∇ y i | approaches 1 sufficiently fast, it can be checked (2) is equivalent to 3 i=1
S∞
∂ |∇ y i | dσ = 16π m(g). ∂ν
(3)
In view of (1) and (3), it is desirable to seek a formula that computes the ADM mass (see [1]) solely in terms of geometric data of the level sets of y i near infinity. In this paper, we derive formulae of this nature. As the level sets of y i are simply coordinate planes, we are thus prompted to compute m(g) on the boundary of large coordinate cubes. A Riemannian 3-manifold (M, g) is called asymptotically flat with a metric falloff rate τ if there exists a coordinate chart {x i }, outside a compact set, in which the metric coefficients satisfies gi j = δi j + O(|x|−τ ), ∂gi j = O(|x|−τ −1 ), ∂∂gi j = O(|x|−τ −2 ).
(4)
The scalar curvature R of g is assumed to be integrable so tha
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