一类初值数据拟局部质量正性的紧凑证明
Compact Proof of the Positivity of Quasi-Local Masses for a class of Initial Data
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中文总结 AI 辅助
本文针对一类初值数据，通过Jang形变和共形约化至零标量曲率，利用Brown-York质量的第二变分，在紧致填充内证明了Wang-Yau质量的严格正性，无需渐近平坦延拓或正质量定理，部分回答了Schoen的问题。
中文摘要 AI 辅助
我们证明了一个纯拟局部的正性定理，针对一类初值数据的Wang--Yau质量，这类数据的Jang形变在保边界共形约化到零标量曲率后，位于严格凸欧氏度量的一个足够小的横迹-无迹（TT）微扰邻域内。我们显式构造了一类非平凡的物理初值数据，其允许的Jang约化实现了这一TT生成的扇区。该论证将Wang--Yau能量约化为所得标量平坦紧致度量的Brown--York质量，加上由Jang形变决定的非负体积项，并通过在欧氏度量处沿横迹-无迹方向计算Brown--York泛函的第二变分来确立严格正性。证明完全局限于紧致填充，既不使用渐近平坦延拓，也不使用正质量定理。这为R. Schoen关于拟局部质量正性的真正拟局部证明的问题提供了一个部分回答。
英文摘要
We prove a purely quasi-local positivity theorem for the Wang--Yau mass for a class of initial data whose Jang deformation, after a boundary-preserving conformal reduction to zero scalar curvature, lies in a sufficiently small transverse--traceless (TT) perturbative neighborhood of a strictly convex Euclidean fill-in.We explicitly construct a nontrivial class of physical initial data whose admissible Jang reductions realize this TT-generated sector. The argument reduces the Wang--Yau energy to the Brown--York mass of the resulting scalar-flat compact metric, together with nonnegative bulk terms determined by the Jang deformation, and establishes strict positivity by computing the second variation of the Brown--York functional at the Euclidean metric in transverse--traceless directions. The proof is entirely confined to the compact fill-in and uses neither an asymptotically flat extension nor the positive mass theorem. This gives a partial answer to a question of R. Schoen concerning a genuinely quasi-local proof of positivity for quasi-local mass.
