Related papers: Existence of static vacuum extensions with prescri…

We introduce the notions of static regular of type (I) and type (II) and show that they are sufficient conditions for local well-posedness of solving asymptotically flat, static vacuum metrics with prescribed Bartnik boundary data. We then…

In our prior work toward Bartnik's static vacuum extension conjecture for near Euclidean boundary data, we establish a sufficient condition, called static regular, and confirm large classes of boundary hypersurfaces are static regular. In…

We obtain existence and local uniqueness of asymptotically flat, static vacuum extensions for Bartnik data on a sphere near the data of a sphere of symmetry in a Schwarzschild manifold.

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…

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…

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…

We investigate the Bartnik stationary extension conjecture, which arises from the definition of the spacetime Bartnik mass for a compact region in a general initial data set satisfying the dominant energy condition. This conjecture posits…

We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean…

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.

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $\Pi$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary…

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…

This paper is a tribute to Robert Bartnik and his work and conjectures on quasi-local mass. We present a framework in which to clearly analyse Bartnik's static vacuum extension conjecture. While we prove that this conjecture is not true in…

In this paper, we give a definition for the Bartnik mass of a domain whose extensions are asymptotically hyperbolic manifolds. With this definition, we show that asymptotically hyperbolic admissible extensions of a domain that achieve the…

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…

We study Cauchy initial data for asymptotically flat, stationary vacuum space-times near space-like infinity. The fall-off behavior of the intrinsic metric and the extrinsic curvature is characterized. We prove that they have an analytic…

The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be…

In this paper, we establish that a four-dimensional static vacuum asymptotically flat spacetime containing a massive particle sphere is isometric to the Schwarzschild spacetime. Our results expand upon existing uniqueness theorems for…

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…

Caustics-envelopes formed by the trajectories of fluid particles-arise in proposed dynamical extensions for shell-crossing singularities occurring in the Einstein-dust system. In this study, a local existence result is established,…

We prove the existence of a large class of asymptotically flat initial data with non-vanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at space-like infinity in terms of powers…