AI 中文总结
研究三维和四维带边界的 Yamabe 方程正解在孤立奇点附近的渐近行为，建立了精确上界与不可去情形下的匹配下界及可去性条件，证明不可去奇点解渐近柱对称，扩展相关理论。
AI 中文摘要
本文研究了在度量不是共形平坦时，带边界的 Yamabe 方程正解在孤立奇点附近的渐近行为。在三维和四维中，我们建立了精确的上界，在不可去情形下，建立了匹配的下界，这也给出了可去性的充要条件。此外，每个具有不可去奇点的解都被证明是渐近柱对称的，而不依赖于 Fowler 型奇异解的全局分类。这些结果还将 Caffarelli-Jin-Sire-Xiong（2014）的平坦半空间理论扩展到非平坦边界几何，并提供了 Marques（2008）和 Xiong-Zhang（2022）发展的内部理论的边界类似物。
英文摘要
This paper studies the asymptotic behavior of positive solutions to the boundary Yamabe equation near an isolated singularity when the metric is not conformally flat. In dimensions 3 and 4, we establishes the sharp upper bound and, in the non-removable case, the matching lower bound, which also gives a necessary and sufficient condition for removability. Moreover, every solution with a non-removable singularity is shown to be asymptotically cylindrically symmetric, without relying on a global classification of Fowler-type singular solutions. These results aslo extend the flat half-space theory of Caffarelli-Jin-Sire-Xiong (2014) to non-flat boundary geometries and provide boundary analogues of the interior theories developed by Marques (2008) and Xiong-Zhang (2022).
