具有正Yamabe度量的Einstein约束方程的远离常平均曲率解
Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics
摘要：
本文利用Lichnerowicz-York共形重标度方法得到了爱因斯坦约束方程解的一些新的存在性结果。平均非本征曲率被认为是一个任意光滑函数，对它的空间导数的大小没有限制，所以它可以是任意远离常数。重新标度的背景度规属于正Yamabe类，并且由重新标度的非本征曲率的无痕横向部分和物质场给出的数据的可自由指定部分被取为足够小，物质能量密度不恒为零。使用拓扑不动点参数和全球壁垒建设，然后我们建立的约束的解决方案的存在性。爱因斯坦约束方程分析的两个最新进展使这一结果成为可能：一种新型的拓扑不动点论点没有小条件的平均外部曲率的空间导数，和一个新的建设的整体超解的哈密顿约束，同样是免费的平均外部曲率的条件。为了清楚起见，我们只给出闭流形上强解的结果。然而，我们的结果也适用于弱解和其他情况，如紧流形的边界;这些推广将出现在其他地方。这里提出的爱因斯坦约束的存在性结果显然是第一个这样的结果，不需要小的平均外部曲率的空间导数的条件。
英文摘要：
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 restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant. The rescaled background metric belongs to the positive Yamabe class, and the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are taken to be sufficiently small, with the matter energy density not identically zero. Using topological fixed-point arguments and global barrier constructions, we then establish existence of solutions to the constraints. Two recent advances in the analysis of the Einstein constraint equations make this result possible: A new type of topological fixed-point argument without smallness conditions on spatial derivatives of the mean extrinsic curvature, and a new construction of global super-solutions for the Hamiltonian constraint that is similarly free of such conditions on the mean extrinsic curvature. For clarity, we present our results only for strong solutions on closed manifolds. However, our results also hold for weak solutions and for other cases such as compact manifolds with boundary; these generalizations will appear elsewhere. The existence results presented here for the Einstein constraints are apparently the first such results that do not require smallness conditions on spatial derivatives of the mean extrinsic curvature.
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