Rough Solutions of the Einstein Constraints on Closed Manifolds without Near-CMC Conditions
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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 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 ∈ Ws,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 Choquet-Bruhat , and in the case p = 2, we reproduce the CMC existence results of Maxwell , 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 allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.