Einstein约束方程共形形式的非唯一和负Yamabe解之间的一种选择
An Alternative Between Non-unique and Negative Yamabe Solutions to the Conformal Formulation of the Einstein Constraint Equations
摘要：
保角方法对于封闭三维流形上的Einstein约束方程的参数化求解是有效的。然而，它仍然没有得到很好的理解;例如，零或负Yamabe度量的共形方程的解的存在性仍然是未知的，没有所谓的“CMC”或“近CMC”假设。第一个没有这种假设的存在结果，称为“远离CMC”的情况，是由Holst，Nagy和Tsogtgerel在2008年获得的正Yamabe度量。然而，他们的结果是基于拓扑参数，因此解决方案的唯一性是未知的。事实上，麦克斯韦在2011年给出的证据表明，在某些情况下，远离CMC的解决方案并不是唯一的。在这篇文章中，我们提供了进一步的洞察力，通过建立一般远离CMC的解决方案的替代定理的类型。对于一个给定的流形M，若度量g（0）具有正的纯量曲率和纯量平坦，且不具有共形Killing场，我们首先证明了存在一个解析的单参数度量族g（z）到g（0），使得R（g（z））= z.使用这一系列的度量和给定的数据（tau，sigma，rho，j），我们形成一个单参数家庭的运营商F（（phi，w），z），其零点满足共形方程。应用Liapnuov-Schmidt约化，通过F（（phi，w），z）线性化消失的临界点，确定了F（（phi，w），z）= 0的解析解曲线.这条曲线的规律性，F（（phi，w），z）的定义，以及Holst等人早期的远离CMC的结果，使我们能够证明以下远离CMC解的替代定理：或者（1）存在z_1 >0，使得z_1参数化共形方程的（正Yamabe）解是非唯一的;或（2）存在z_2 < 0使得z_2参数化共形方程存在（负Yamabe）解。
英文摘要：
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 zero or negative Yamabe metrics is still unknown without the so-called ``CMC'' or ``near-CMC'' assumptions. The first existence results without such assumptions, termed the ``far-from-CMC'' case, were obtained by Holst, Nagy, and Tsogtgerel in 2008 for positive Yamabe metrics. However, their results are based on topological arguments, and as a result solution uniqueness is not known. Indeed, Maxwell gave evidence in 2011 that far-from-CMC solutions are not unique in certain cases. In this article, we provide further insight by establishing a type of alternative theorem for general far-from-CMC solutions. For a given manifold M that admits a metric of positive scalar curvature and scalar flat metric g(0) with no conformal Killing fields, we first prove existence of an analytic, one-parameter family of metrics g(z) through g(0) such that R(g(z)) = z. Using this family of metrics and given data (tau,sigma,rho,j), we form a one-parameter family of operators F((phi,w),z) whose zeros satisfy the conformal equations. Applying Liapnuov-Schmidt reduction, we determine an analytic solution curve for F((phi,w),z) = 0 through a critical point where the linearization of F((phi,w),z) vanishes. The regularity of this curve, the definition of F((phi,w),z), and the earlier far-from-CMC results of Holst et al. allow us to then prove the following alternative theorem for far-from-CMC solutions: either (1) there exists a z_1 >0 such that (positive Yamabe) solutions to the z_1-parameterized conformal equations are non-unique; or (2) there exists z_2 < 0 such that (negative Yamabe) solutions to the z_2-parameterized conformal equations exist.
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