arXiv research

Existence proved for static vacuum extensions near Schwarzschild spheres.

Extends static vacuum metrics with specific boundary conditions.

Paper defines Bartnik mass for hyperbolic extensions and proves staticity.

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…

Proves existence of static vacuum metrics with specific boundary data.

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 $\bar{B}_1$ that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists …

Paper analyzes Bartnik's quasi-local mass conjectures and their validity.

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 …

Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.

Estimates mass of static vacuum metrics with small Bartnik data.

Maximizes capacity of extensions with fixed boundary data.

We solve Bartnik's stationary extension problem near Schwarzschild spheres.

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…

Given a Riemannian 3-ball $(\bar B, g)$ of non-negative scalar curvature, Bartnik conjectured that $(\bar B, g)$ admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass, and that such a mass-minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined b…

New static vacuum metrics confirmed for near Euclidean boundary data.

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…

Paper proves rigidity of static manifolds and applies to metric extensions.

We prove that given any smooth metric $γ$ and smooth positive function $H$ on $S^{2}$, 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 $M = {\mathbb R}^{3} \setminus B^{3}$, such that the induced metric and mean curvature of $…

We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary.

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…

Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface $Σ$, we observe that …

We show that if an asymptotically flat manifold with horizon boundary admits a global static potential, then the static potential must be zero on the boundary. We also show that if an asymptotically flat manifold with horizon boundary admits an unbounded static potential in the exterior region, then the manifold must c…

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…

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…

Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds $(Σ\cong \mathbb{S}^2,g)$, with $g$ satisfying $λ_1 = λ_1(-Δ_g + K(g))>0$, where $λ_1$ is the first eigenvalue of the operator $-Δ_g+K(g)$ and $K(g)$ is the Gaussian curvature of $g$, with control on t…

Constructs initial data for Einstein equations and estimates Bartnik mass outside time-symmetry.

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $Π$ in $\mathbb E$, 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…

Bartnik mass is positive and non-decreasing for black holes

Proves critical points of ADM mass correspond to specific initial data sets.

This paper constructs charged Riemannian manifolds to test Penrose inequality.

Consider a triple of "Bartnik data" $(Σ, γ,H)$, where $Σ$ is a topological 2-sphere with Riemannian metric $γ$ and positive function $H$. We view Bartnik data as a boundary condition for the problem of finding a compact Riemannian 3-manifold $(Ω,g)$ of nonnegative scalar curvature whose boundary is isometric to $(Σ,γ)$…

We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the ex…

We construct asymptotically flat, scalar flat extensions of Bartnik data $(Σ, γ, H)$, where $γ$ is a metric of positive Gauss curvature on a two-sphere $Σ$, and $H$ is a function that is either positive or identically zero on $Σ$, such that the mass of the extension can be made arbitrarily close to the half area radius…

Provides an overview of Bartnik's quasi-local mass.

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 $Σ_t$ evolving in an initial data set {with the dominant energy condit…

This paper surveys recent progress on issues related to the Bartnik quasi-local mass $m_B$. In addition, we formulate a number of new problems and conjectures regarding foundational properties of the mass $m_B$. This work is dedicated with pleasure to Robert Bartnik in honor of his 60th birthday.

New insights into Bartnik mass from improvability of dominant energy scalar.

Let $g$ be a metric on the $2$-sphere $\mathbb{S}^2$ with positive Gaussian curvature and $H$ be a positive constant. Under suitable conditions on $(g, H)$, we construct smooth, asymptotically flat $3$-manifolds $M$ with non-negative scalar curvature, with outer-minimizing boundary isometric to $(\mathbb{S}^2, g)$ and …

Study metrics with specific spectral properties to compute Bartnik and Bartnik-Bray masses efficiently.

Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor $W$ to the setting of smooth metric measure space…

Establishes a version of Bartnik's conjecture for Lorentzian length spaces.

Adapts Bartnik method to Hilbert manifold structure for vacuum constraint equations.

New bounds on Bartnik mass for surfaces with non-negative first eigenvalue.

New order defined for conformal classes, impacts Bartnik's conjecture.

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…

Simple proof for sphere mass calculation.

Estimates Bartnik mass for metrics with nonnegative Gauss 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.