# Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions - General Relativity and Quantum Cosmology

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Abstract: We consider the conformal decomposition of Einstein-s constraint equationsintroduced by Lichnerowicz and York, on a closed manifold. We establishexistence of non-CMC weak solutions using a combination of a priori estimatesfor the individual Hamiltonian and momentum constraints, barrier constructionsfor the Hamiltonian constraint, and topological fixed-point arguments. Animportant new feature of these results is the absense of the near-CMCassumption when the rescaled background metric is in the positive Yamabe class,if the freely specifiable part of the data given by the matter fields ifpresent and the traceless-transverse part of the rescaled extrinsic curvatureare taken to be sufficiently small. In this case, the mean extrinsic curvaturecan be taken to be an arbitrary smooth function without restrictions on thesize of its spatial derivatives, giving what are apparently the first non-CMCexistence results without the near-CMC assumption. Standard bootstrappingarguments to increase the regularity of the conformal factor are blocked by theuse of a weak background metric. In the CMC case, we recover Maxwell-s roughsolution results as a special case. Our results extend the 1996 non-CMC resultof Isenberg and Moncrief in three ways: 1 the near-CMC assumption is removedin the case of the positive Yamabe class; 2 regularity is extended down tothe maximum allowed by the background metric and the matter; and 3 the resultholds for all three Yamabe classes. This last extension was also accomplishedrecently by Allen, Clausen and Isenberg, although their result is restricted tothe near-CMC case and to smoother background metrics and data.

Author: ** Michael Holst, Gabriel Nagy, Gantumur Tsogtgerel**

Source: https://arxiv.org/