Related papers: Existence of static vacuum extensions with prescri…
We apply the method of matched asymptotic expansions to analyse whether cosmological variations in physical `constants' and scalar fields are detectable, locally, on the surface of local gravitationally bound systems such as planets and…
A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…
We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any…
In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant $(n-2)$-curvature and a prescribed asymptotic boundary at infinity. Previously, the existence was known only for a restricted…
In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds $(M^{n}, g)$ with boundary and with dimension $ n < 8$ that was establishedby McCormick. First, we show that any asymptotically flat static…
We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as `null data'. These are given by sequences of symmetric trace…
Inspired by R. Bartnik's mass minimization problem in general relativity, we investigate a dual problem of maximizing the capacity among asymptotically flat extensions (with nonnegative scalar curvature) of some fixed two-dimensional…
This paper examines the issue of the existence and nature of time-like geodesics in asymptotically flat spacetimes and proposes a novel generalized topological criterion for the existence of time-like geodesics. Its validity is proved using…
We investigate lower asymptotic bounds of number variances for invariant locally square-integrable random measures on Euclidean and real hyperbolic spaces. In the Euclidean case we show that there are subsequences of radii for which the…
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…
We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…
Bartnik's quasi-local mass is a functional on Bartnik data $(\mathbb S^2,\gamma,H,P,\omega^\perp)$, consisting of a metric $\gamma$, scalar functions $H$ and $P$, and a 1-form $\omega^\perp$ on the $2$-sphere $\mathbb S^2$. We construct…
We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null…
Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space.…
We construct asymptotically flat, scalar flat extensions of Bartnik data $(\Sigma, \gamma, H)$, where $\gamma$ is a metric of positive Gauss curvature on a two-sphere $\Sigma$, and $H$ is a function that is either positive or identically…
We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the…
Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether…
Systematic numerical investigations of the asymptotics of near Schwarzschild vacuum initial data sets is carried out by inspecting solutions to the parabolic-hyperbolic and to the algebraic-hyperbolic forms of the constraints, respectively.…
Given asymptotically flat initial data on M^3 for the vacuum Einstein field equation, and given a bounded domain in M, we construct solutions of the vacuum constraint equations which agree with the original data inside the given domain, and…
The classification of certain class of static solutions for the Einstein-Gauss-Bonnet theory in vacuum is presented. The spacelike section of the class of metrics under consideration is a warped product of the real line with a nontrivial…