Related papers: Progress on nonuniqueness of solutions to the vacu…
In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe…
In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer's fixed point theorem (known methods use Schauder's fixed point theorem) while the second one uses the concept…
A closed explicit representation of the vacuum Einstein equations in terms of components of curvature 2-forms is given. The discussion is restricted to the case of non-vanishing cubic invariant of conformal curvature spinor. The complete…
In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15, 16, 24], namely existence when the so called TT-tensor…
We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of…
We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new…
In this article we develop some new existence results for the Einstein constraint equations using the Lichnerowicz-York conformal rescaling method. The mean extrinsic curvature is taken to be an arbitrary smooth function without…
We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the Einstein constraint…
Given a conformally variational scalar Riemannian invariant $I$, we identify a sufficient condition for a compact Riemannian manifold to admit finite regular coverings with many nonhomothetic conformal rescalings with $I$ constant. We also…
Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $\sigma \in W^{1,p}$ and a positive function $\tau \in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations…
We give some uniform estimates for constant mean curvature solutions of the conformal vacuum Einstein constraint equations on compact manifolds. Existence of those solutions was given in a paper by J. Isenberg.
In this note we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the TT-tensor is small in $L^2$.
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…
We propose a revised formulation of General Relativity for cosmological settings, in which the Einstein constant varies with the energy density of the Universe. We demonstrate that this modification has only phenomenological impact of…
When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal…
The approach, referred to as "monodromy transform", provides some general base for solution of all known integrable space - time symmetry reductions of Einstein equations for the case of pure vacuum gravitational fields, in the presence of…
The Einstein static universe has played a central role in a number of emergent scenarios recently put forward to deal with the singular origin of the standard cosmological model. Here we study the existence and stability of the Einstein…
We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a…
All Lorentz invariant solutions of vacuum Einstein's equations are found. It is proved that these solutions describe space-times of constant curvature.
In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very…