A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Refines geometric center of mass analysis for Einstein field equations.
Paper introduces new center of mass for flat manifolds.
The classical notion of center of mass for an isolated system in general relativity is derived from the Hamiltonian formulation and represented by a flux integral at infinity. In contrast to mass and linear momentum which are well-defined for asymptotically flat manifolds, center of mass and angular momentum seem less …
We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using Ricci tensor are the sa…
Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.
We calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau \cite{CWY} for a family of spacelike two-spheres approaching future null infinity in an asymptotically flat spacetime admitting a Bondi-Sachs expansion. Our result complements earlier work of Chen-Wang-Yau \cite{CWY2…
Study on Poncelet polygons' centers and circumcenters in various geometries.
Study mass and center of mass in flat 3-manifolds, proving existence of foliations.
Formulae track evolution of angular momentum and center of mass at null infinity.
It is shown by several authors going back to Huisken-Yau that asymptotically Schwarzschildean time-slices possess a unique foliation by stable constant mean curvature (CMC) spheres defining the so-called CMC center of mass. We analyze how the leaves of this foliation evolve in time under the Einstein equations. More pr…
The (relativistic) center of mass of an asymptotically flat Riemannian manifold is often defined by certain surface integral expressions evaluated along a foliation of the manifold near infinity, e. g. by Arnowitt, Deser, and Misner (ADM). There are also what we call 'abstract' definitions of the center of mass in term…
We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minko…
We will discuss existence of center of mass on asymptotically Schwarzschild manifold defined by Huisken-Yau and Corvino-Schoen. Conditions of existence and examples on non existence are given.
The center of mass in General Relativity is hard to define due to coordinate freedom.
This thesis discusses the Newtonian limit of General Relativity for static isolated systems with compactly supported matter. We call these systems "geometrostatic" to underline their geometric nature. We introduce new quasi-local notions of mass and center of mass that can be read off locally in the vicinity of the mat…
In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this CMC-foliation. Rigger and Neves-Tian showed in 2004 and 2009/10 analogous existence and uni…
New inequality on sphere generalizes circle inequality.
New definitions of conserved quantities at null infinity resolve ambiguities in general relativity.
We present an iterative technique for finding zeroes of vector fields on Riemannian manifolds. As a special case we obtain a ``nonlinear averaging algorithm'' that computes the centroid of a mass distribution supported in a set of small enough diameter D in a Riemannian manifold M. We estimate the convergence rate of o…
We present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems. We then explain the pseudo-Newtonian characte…
The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.
We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliat…
We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a …
Given a sphere with Bartnik data close to that of a round sphere in Euclidean 3-space, we compute its Bartnik-Bray outer mass to first order in the data's deviation from the standard sphere. The Hawking mass gives a well-known lower bound, and an upper bound is obtained by estimating the mass of a static vacuum extensi…
We propose a definition of center of mass for asymptotically flat manifolds satisfying Regge-Teitelboim condition at infinity. This definition has a coordinate-free expression and natural properties. Furthermore, we prove that our definition is consistent both with the one proposed by Corvino and Schoen and another by …
The Riemannian center of mass was constructed in [GrKa] (1973). In [GKR1, GKR2, Gr, Ka, BuKa] (1974-1981) it was successfully applied with more refined estimates. Probably in 1990 someone renamed it without justification into karcher mean and references to the older papers were omitted by those using the new name. As a…
We consider the two body problem with central interaction on two point homogeneous spaces from point of view of the invariant differential operators theory. The representation of the two particle Hamiltonian in terms of the radial differential operator and invariant operators on the symmetry group is found. The connect…
In 1996, Huisken-Yau showed that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed CMC-surfaces if it is asymptotically equal to the (spatial) Schwarzschild solution and has positive mass. Their assumptions were later weakened by Metzger, Huang, Eichmair-Metzger and oth…
We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisym…
In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previo…
In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken \cite{Huisken}. We will…
Characterizes curves for minimal surfaces in de Sitter space.
The paper proves the existence of stable spheres in asymptotically flat 3-manifolds.
Defines circumcenter of mass for polytopes without triangulation.
The purpose of this article is to \begin{enumerate} \item define the -fold center of mass arrangement for points in the plane, \item give elementary properties of and \item give consequences concerning the space of distinct points in the plane, no four of which are the vertices of …
We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [26] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group…
This paper proves a canonical foliation on null infinity for Kerr-like black holes.
New positive mass theorem for hyperbolic 3-manifolds using Green functions.
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…
The paper establishes preferred coordinates for AE 3-manifolds, improving ADM center of mass convergence.
We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asympto…
Quadratic memory is essential for optimal convex optimization queries.
In the first half of this article, we survey the new quasi-local and total angular momentum and center of mass defined in [9] and summarize the important properties of these definitions. To compute these conserved quantities involves solving a nonlinear PDE system (the optimal isometric embedding equation), which is ra…
We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex." Specifically, a set of weights is mapped to the Riemannian center of mass for the…
These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein's time. I shall explain these difficulties and progresses that have been made. In particular, I shall i…
The study extends conserved quantities theory to non-compact boundary initial data sets.
Improved Sobolev mappings in Carnot groups with weaker assumptions.
Constructs constant spacetime mean curvature surfaces for hyperboloidal initial data sets.