In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold (Ω,γ) with boundary. In one case, the extension is taken to be a manifold without boundary in which (Ω,γ) embeds isometrically, and in the other case the extension is taken to be a m…
Adapts Bartnik method to Hilbert manifold structure for vacuum constraint equations.
problem Vacuum constraint equations on compact manifolds of any dimension ≥ 3.
method Adapts Bartnik method to provide Hilbert manifold structure.
result Fibers of scalar curvature and constraint operator are Hilbert submanifolds.
Constructs initial data for Einstein equations and estimates Bartnik mass outside time-symmetry.
problem Estimating Bartnik mass outside time-symmetry.
method Constructs initial data for Einstein equations and connects Bartnik data to time-symmetric data.
result Obtains estimates for the Bartnik mass outside of time-symmetry.
Paper analyzes Bartnik's quasi-local mass conjectures and their validity.
problem Understanding the validity of Bartnik's quasi-local mass conjectures.
method Developed a framework to analyze Bartnik's static vacuum extension conjecture.
result Proved the static vacuum extension conjecture is not true in general.
Study metrics with specific spectral properties to compute Bartnik and Bartnik-Bray masses efficiently.
problem Compute Bartnik and Bartnik-Bray masses efficiently for metrics with specific spectral properties.
method Spectral generalization of positive scalar curvature, applying Codá Marques's path-connectedness theorem, and efficient constructions for scalar-nonnegative fill-in problem.
result Compute Bartnik and Bartnik-Bray masses efficiently for metrics with -Δ + kR ≥ 0.
Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mea…
We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed geometric or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasi-local mass. A variational characterization…
Provides an overview of Bartnik's quasi-local mass.
problem Understanding Bartnik's quasi-local mass.
method Surveys results on Bartnik's quasi-local mass.
result Serves as an entry point and quick reference.
It is conjectured that the full (spacetime) Bartnik mass of a surface Σ is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by Σ. Assuming this holds true for a 1-parameter family of surfaces Σt evolving in an initial data set {with the dominant energy condit…
Motivated by problems related to quasi-local mass in general relativity, we study the static metric extension conjecture proposed by R. Bartnik \cite{Bartnik_energy}. We show that, for any metric on Bˉ1 that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists …
Extends static vacuum metrics with specific boundary conditions.
problem Proving the existence of static vacuum metrics with prescribed boundary data.
method Introducing static regular types (I) and (II), showing local well-posedness, and confirming Bartnik's conjecture.
result Confirms Bartnik's static vacuum extension conjecture for a broad range of boundary conditions.
Existence proved for static vacuum extensions near Schwarzschild spheres.
problem Proving existence of static vacuum extensions near Schwarzschild spheres.
method Existence and local uniqueness of static vacuum extensions for Bartnik data on a sphere near a Schwarzschild sphere.
result Existence of static vacuum extensions near Schwarzschild spheres.
This paper surveys recent progress on issues related to the Bartnik quasi-local mass mB. In addition, we formulate a number of new problems and conjectures regarding foundational properties of the mass mB. This work is dedicated with pleasure to Robert Bartnik in honor of his 60th birthday.
Estimates mass of static vacuum metrics with small Bartnik data.
problem Estimating mass of static vacuum metrics with small perturbations.
method Second-order mass estimation using Bartnik data.
result New upper bound on Bartnik mass to fifth order.
New insights into Bartnik mass from improvability of dominant energy scalar.
problem Characterizing Bartnik mass minimizing initial data sets.
method Introducing improvability concept, proving non-improvability consequences, and analyzing pp-wave counterexamples.
result Bartnik mass minimizing initial data sets are characterized, advancing conjectures.
We establish a moduli space E of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map Π in E, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map Π is Fredholm by showing that the stationary vacuum equations (combined with p…
The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik…
Proves existence of static vacuum metrics with specific boundary data.
problem Existence of static vacuum metrics with prescribed boundary data.
method Proves existence and local uniqueness of static vacuum metrics close to the Euclidean metric.
result Existence of static vacuum metrics with prescribed Bartnik boundary data.
Establishes a version of Bartnik's conjecture for Lorentzian length spaces.
problem Proving Bartnik's conjecture for Lorentzian length spaces.
method Using timelike completeness and non-negative timelike curvature bounds, the causal boundary is shown to be a single point.
result A globally hyperbolic Lorentzian length space splits as a metric Lorentzian product.
Bartnik mass is positive and non-decreasing for black holes
problem Quasilocal mass for black holes
method Defining a Bartnik mass and proving positivity and monotonicity
result Positive and non-decreasing Bartnik mass for black holes
We investigate Bartnik's static metric extension conjecture under the additional assumption of axisymmetry of both the given Bartnik data and the desired static extensions. To do so, we suggest a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum e…
Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.
problem Proving well-posedness for the Bartnik static extension problem near Schwarzschild spheres.
method Introduced a geodesic gauge to formulate governing equations as coupled elliptic and transport equations; used Bochner-measurable functions for transport equations.
result Established local well-posedness for arbitrary Bartnik data near Schwarzschild spheres, including those with small mean curvature.
Paper defines Bartnik mass for hyperbolic extensions and proves staticity.
problem Defining and proving staticity of asymptotically hyperbolic minimal mass extensions.
method Definition of Bartnik mass, construction of metrics, one-parameter family analysis.
result Static potential for asymptotically hyperbolic admissible extensions achieving Bartnik mass.
The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to com…
New bounds on Bartnik mass for surfaces with non-negative first eigenvalue.
problem Bounding Bartnik mass for surfaces with spectral non-negativity condition.
method Proving upper bound on Bartnik mass using spectral non-negativity condition.
result Bounded above by √(|S²|_g/16π) under spectral non-negativity.
We solve Bartnik's stationary extension problem near Schwarzschild spheres.
problem Existence and uniqueness of asymptotically flat stationary vacuum spacetimes.
method Developed a double geodesic gauge, reducing equations to elliptic and transport-type problems.
result Local well-posedness for Bartnik stationary metric extension problem near Schwarzschild spheres.
New order defined for conformal classes, impacts Bartnik's conjecture.
problem Bartnik's conjecture and its implications under different energy conditions.
method Defined a new order on conformal classes and analyzed implications under null energy condition.
result The null energy condition can lead to future complete metrics in any dimension.
We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik …
Researchers prove a 30-year-old cosmological conjecture about spacetime.
problem The rigidity of the cosmological Hawking--Penrose singularity theorem.
method Combining global viscosity solutions and elliptic approaches.
result A timelike geodesically complete spacetime splits isometrically as a Lorentzian product.
Proves critical points of ADM mass correspond to specific initial data sets.
problem Finding initial data sets with fixed Bartnik boundary data.
method Proves existence of critical points on a Banach manifold.
result Critical points of ADM mass correspond to initial data sets with generalized Killing vector fields.
Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds (Σ≅S2,g), with g satisfying λ1=λ1(−Δg+K(g))>0, where λ1 is the first eigenvalue of the operator −Δg+K(g) and K(g) is the Gaussian curvature of g, with control on t…
Estimates Bartnik mass for metrics with nonnegative Gauss curvature.
problem Estimating Bartnik mass for specific metric configurations.
method Using area, total mean curvature, and a metric roundness measure.
result Estimate approaches sharp value for round spheres.
Maximizes capacity of extensions with fixed boundary data.
problem Maximizing the capacity of extensions with nonnegative scalar curvature.
method Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions.
result Derives variational condition for maximal capacity extensions and proves they have constant scalar curvature.
Generalising a proof by Bartnik in the asymptotically Euclidean case, we give an elementary proof of positivity of the hyperbolic mass near the hyperbolic space. It is a pleasure to dedicate this work to Robert Bartnik on the occasion of his 60th birthday.
Upper bounds on Bartnik mass for non-negatively curved spheres.
problem Bounding Bartnik mass for non-negatively curved spheres.
method Establishing upper bounds using non-negative Gauss curvature.
result Upper bounds on Bartnik mass approach Hawking mass under certain conditions.
The paper shows how to create Schwarzschild initial data with degenerate apparent horizons.
problem Creating Schwarzschild initial data with degenerate apparent horizons.
method Modifying the construction of Mantoulidis-Schoen to handle the degenerate case.
result The first eigenvalue of the operator Lg must be zero for Schwarzschild initial data with degenerate apparent horizons. New formula shows how causal vectors relate to mass-minimizing data.
problem Understanding mass-minimizing initial data sets and their geometry.
method Developed a new monotonicity formula for causal Killing vectors.
result Established strong maximum principles for the Lorentzian length.
Quite a number of distinct versions of Bartnik's definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary cond…
In this paper we characterize the intrinsic geometry of apparent horizons (outermost marginally outer trapped surfaces) in asymptotically flat spacetimes; that is, the Riemannian metrics on the two sphere which can arise. Furthermore we determine the minimal ADM mass of a spacetime containing such an apparent horizon. …
New vacuum spacetimes without CMC Cauchy surfaces found.
problem Finding vacuum cosmological spacetimes without CMC Cauchy surfaces.
method Extended construction of [6] using spatial topologies M#M. result Obtained a large class of vacuum cosmological spacetimes.
The Positive Mass Theorem for special singular initial data.
problem Proving the positive mass theorem for data with a codimension one singularity.
method Using asymptotically flat spin initial data sets with matching Bartnik data condition involving spacetime rotations.
result Established a spacetime positive mass theorem and rigidity statement.
Inspired by the results in a recent paper by G. Galloway and C. Vega (see arXiv:1712.00785), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field on a globally hyperbolic spacetime with compact Cauchy hypersurfaces, especially in connection with the so-called B…
We prove that given any smooth metric γ and smooth positive function H on S2, there is a constant λ>0, depending on (γ,H), and an asymptotically flat solution (M,g,u) of the static vacuum Einstein equations on M=R3∖B3, such that the induced metric and mean curvature of $…
The paper proves nonexistence of NNSC cobordism for Bartnik data under certain conditions.
problem Proving nonexistence of NNSC cobordism for Bartnik data (Σ1n−1,γ1,H1) and (Σ2n−1,γ2,H2). method Analyzing metrics γ1 and γ2 on Sn−1 with fixed mean curvature H1 and large enough H2 to prove nonexistence of NNSC cobordism. result Proves nonexistence of NNSC cobordism for Bartnik data under specific conditions.
Proves spacetime positive mass theorem with corners.
problem Proving a positive mass theorem for spacetime with corners.
method Deformation theorem with corner conditions, asymptotically flat initial data.
result Exterior end satisfies E≥∣P∣ in every dimension n≥3. Rigidity results for Hawking mass in curved spaces with bounds on Bartnik capacity.
problem Rigidity of surfaces in curved spaces with mass bounds.
method Analyzing Hawking mass and applying rigidity results to specific geometric settings.
result Explicit lower bounds on Hawking and Bartnik masses in non-flat spaces.
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…
This paper constructs charged Riemannian manifolds to test Penrose inequality.
problem Testing the Riemannian Penrose Inequality with charged manifolds.
method Constructing asymptotically hyperbolic or Euclidean extensions with electric charge.
result Suggests instability of the generalized Riemannian Penrose Inequality.