标题： 爱因斯坦约束方程共形形式的非唯一和负Yamabe解之间的另一种选择 显示英文标题

标题： An Alternative Between Non-unique and Negative Yamabe Solutions to the Conformal Formulation of the Einstein Constraint Equations

摘要： 共形方法已被证明在对闭合3流形上的爱因斯坦约束方程的解进行参数化方面是有效的。 然而，它仍然不是完全理解的；例如，在所谓的“CMC”或“near-CMC”假设之外，零或负的Yamabe度量的共形方程的解的存在性仍然是未知的。 没有这些假设的第一个存在结果，称为“far-from-CMC”情况，是由Holst、Nagy和Tsogtgerel在2008年针对正的Yamabe度量得到的。 然而，他们的结果基于拓扑论证，因此解的唯一性尚不清楚。 事实上，Maxwell在2011年提供了证据表明，在某些情况下，far-from-CMC的解不是唯一的。 在本文中，我们通过为一般的far-from-CMC解建立一种替代定理来提供进一步的见解。 对于一个允许具有正标量曲率的度量和无共形Killing场的标量平坦度量g(0)的流形M，我们首先证明存在一个通过g(0)的解析的一参数度量族g(z)，使得R(g(z)) = z。 使用这个度量族和给定的数据(tau,sigma,rho,j)，我们构造一个一参数算子F((phi,w),z)族，其零点满足共形方程。 应用Liapnuov-Schmidt约简，我们在一个线性化F((phi,w),z)消失的关键点处确定了F((phi,w),z) = 0的解析解曲线。 这条曲线的正则性、F((phi,w),z)的定义以及Holst等人之前的far-from-CMC结果使我们能够随后证明以下关于far-from-CMC解的替代定理：要么(1)存在一个z_1 > 0，使得z_1参数化的共形方程的正Yamabe解是非唯一的；要么(2)存在一个z_2 < 0，使得z_2参数化的共形方程的负Yamabe解存在。 显示英文摘要

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.