Related papers: Nonexistence and Nonuniqueness Results for Solutio…
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 $L^2$.
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…
The conformal method has been effective for parametrizing solutions to the Einstein constraint equations on closed 3-manifolds. However, it is still not well-understood; for example, existence of solutions to the conformal equations for…
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…
In this article, we make a generalization of classical fixed point theorems by using the concept of half-continuity and then apply it to improve the nonuniqueness result for solutions to the vacuum Einstein conformal equations shown by the…
It is well-known that solutions to the conformal formulation of the Einstein constraint equations are unique in the cases of constant mean curvature (CMC) and near constant mean curvature (near-CMC). However, the new far-from-constant mean…
In this paper, we prove a far-from-CMC result similar to the ones obtained by Holst, Nagy, Tsogtgerel and Maxwell for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe…
We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of a priori estimates for the…
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…
The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic…
We extend the study of the vacuum Einstein constraint equations on manifolds with ends of cylindrical type initiated by Chru\'sciel and Mazzeo by finding a class of solutions to the fully coupled system on such manifolds. We show that given…
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…
In this work, we use the global analysis and degree-theoretic methods introduced by Smale to study the existence and multiplicity of solutions of the vacuum Einstein constraint equations given by the conformal method of…
In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have…
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…
In this paper we analyse semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically Euclidean (AE)…
We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by…
The conformal formulation of the Einstein constraint equations has been studied intensively since the modern version of the conformal method was first pub- lished in the early 1970s. Proofs of existence and uniqueness of solutions were…
In this note we examine some recently proposed solutions of the linearized vacuum Einstein equations. We show that such solutions are {\it not} symmetries of the Einstein equations, because of a crucial integrability condition.
We construct low regularity solutions of the vacuum Einstein constraint equations on compact manifolds. On 3-manifolds we obtain solutions with metrics in $H^s$ where $s>3/2$. The constant mean curvature (CMC) conformal method leads to a…