Related papers: Measuring Mass via Coordinate Cubes

We express the $q$-th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total $(2q)$-th mean curvature and the integral, over the entire manifold, of the inner product between…

The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using…

We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the $\sigma_k$ curvature vanishes…

We give an account of some recent development that connects the concept of mass in general relativity to the geometry of large Riemannian polyhedra, in the setting of both asymptotically flat and asymptotically hyperbolic manifolds.

We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the…

It is shown that the mass of an asymptotically flat manifold with a noncompact boundary can be computed in terms of limiting surface integrals involving the Einstein tensor of the interior metric and the Newton tensor attached to the second…

We show that the concept of the ADM mass in general relativity can be understood as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large Riemannian polyhedra. We also express the…

We show how to define curvature as a measure using the Gauss-Bonnet Theorem on a family of singular surfaces obtained by gluing together smooth surfaces along boundary curves. We find an explicit formula for the curvature measure as a sum…

We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and…

Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric identities which relate the scalar curvature of Riemannian 3-manifolds to global invariants in terms of harmonic functions. These quantitative formulas are useful…

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay…

On a closed manifold, consider the space of all Riemannian metrics for which -Delta + kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature…

On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was…

We construct a model that allows us to determine the three neutrino masses and the mass matrix directly from the experimental mass squared differences delta atm and delta sol, using the assumptions of rational hierarchy (m1/m2 = m2/m3) and…

Any compact manifold with positive scalar curvature has an associated asymptotically flat metric constructed using the Green's function of the conformal Laplacian, and the mass of this metric is an important geometric invariant. An explicit…

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in…

In this survey article we review several results on the curvature of semi-Riemannian metrics which are motivated by the positive mass theorem. The main themes are estimates of the Riemann tensor of an asymptotically flat manifold and the…

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat…

We study harmonic maps from a 3-manifold with boundary to $\mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $\pi / 2$. Furthermore we give some applications to mapping torus…

We give a relationship that yields an effective geometric way of evaluating mean curvature of surfaces. The approach is reminiscent of the Gauss's contour based evaluation of intrinsic curvature. The presented formula may have a number of…