Rough Solutions of the Einstein Constraints on Closed Manifolds without Near-CMC Conditions

Abstract

Abstract: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 individual Hamiltonian and momentum constraints, barrier constructions and fixed-point techniques for the Hamiltonian constraint, Riesz-Schauder theory for the momentum constraint, together with a topological fixed-point argument for the coupled system. Although we present general existence results for non-CMC weak solutions when the rescaled background metric is in any of the three Yamabe classes, an important new feature of the results we present for the positive Yamabe class is the absence of the near-CMC assumption, if the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small, and if the energy density of matter is not identically zero. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant, giving what is apparently the first existence results for non-CMC solutions without the near-CMC assumption. Using a coupled topological fixed-point argument that avoids near-CMC conditions, we establish existence of coupled non-CMC weak solutions with (positive) conformal factor φ ∈ W s, p , where p ∈ (1, ∞) and s( p) ∈ (1 + 3/ p, ∞). In the CMC case, the regularity can be reduced to p ∈ (1, ∞) and s( p) ∈ (3/ p, ∞) ∩ [1, ∞). In the case of s = 2, we reproduce the CMC existence results of , and in the case p = 2, we reproduce the CMC existence results of Maxwell [33], but with a proof that goes through the same analysis framework that we use to obtain the non-CMC results. The non-CMC results on closed manifolds here extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum

Far-from-Constant Mean Curvature Solutions of Einstein’s Constraint Equations with Positive Yamabe Metrics

“…We state only the results for strong solutions, recovering previous results in [2,4]. Generalizations allowing weaker differentiability conditions on the coefficients appear in [7,8].…”

“…We have generalized this result in [7,8], allowing weaker coefficient differentiability, giving existence of solutions down to w a ∈ W 1,p , with real number p 2. The proof in [8] is based on RieszSchauder theory for compact operators [13]. The case of compact manifold M with boundary is analyzed in [7].…”

“…These two results allow us to establish existence of non-CMC solutions for conformal background metrics in the positive Yamabe class, with the freely specifiable part of the data given by the traceless transverse part of the rescaled extrinsic curvature and the matter fields sufficiently small, and with the matter energy density not identically zero. We only state the main results and give the ideas of the proofs; detailed proofs may be found in [8] for closed manifolds and in [7] for compact manifolds with boundary. Our results here and in [7,8] reduce the remaining open questions of existence of non-CMC solutions without near-CMC conditions to two basic open questions: (1) Existence of global super-solutions for background metrics in the nonpositive Yamabe classes and for large data; and (2) existence of global sub-solutions for background metrics in the positive Yamabe class in vacuum.…”

“…We only state the main results and give the ideas of the proofs; detailed proofs may be found in [8] for closed manifolds and in [7] for compact manifolds with boundary. Our results here and in [7,8] reduce the remaining open questions of existence of non-CMC solutions without near-CMC conditions to two basic open questions: (1) Existence of global super-solutions for background metrics in the nonpositive Yamabe classes and for large data; and (2) existence of global sub-solutions for background metrics in the positive Yamabe class in vacuum.…”

“…is a topologically closed subset of L p , 1 p ∞ (see [8]). Global supersolutions are difficult to find as a consequence of the non-negativity of a w and its estimate (8), together with the limit (11).…”