发表机构
Stony Brook University（石溪大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明正Yamabe--Escobar问题在3≤n≤14时紧致性猜想成立，并构造n≥15的非紧致序列，确定各维度阈值，同时验证Weyl--脐性消失猜想并给出反例。
AI 中文摘要
我们研究Yamabe--Escobar方程，该方程设定正的常数标量曲率和极小边界。紧致性猜想询问：是否共形等价于圆半球是其正解集合紧致性的唯一障碍。在本文中，我们证明该猜想在维度$3\leq n\leq 14$时成立。我们通过构造$n\geq 15$中非脐且非局部共形平坦的半球度量例子，其中存在非紧致解序列，表明该维度阈值$n_*=14$是尖锐的。对于局部共形平坦度量的特殊类别，我们同样做到这一点，其阈值为$n_*^{LCF}=18$，对于具有脐边界的度量，阈值为$n_*^{umb}=20$。在所有三种情形中，我们还证明Weyl--脐性消失猜想在$4\leq n\leq n_*$时成立，并通过扰动非紧致性例子在$n>n_*$时构造反例。据作者所知，这些是正标量曲率、极小边界情形下Yamabe--Escobar方程在圆半球共形类之外的首次局部共形平坦非紧致性构造。
英文摘要
We study the Yamabe--Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dimensions $3\leq n\leq 14$. We show that this dimensional threshold $n_*=14$ is sharp by constructing in $n\geq 15$ nonumbilic and non-locally-conformally-flat examples of metrics in the hemisphere for which a noncompact sequence of solutions exists. We do the same for the special classes of locally conformally flat metrics, $n_*^{LCF}=18$, and the metrics with umbilic boundary, $n_*^{umb} =20$. In all three settings we also prove that the Weyl--umbilicity vanishing conjecture holds for $4\leq n\leq n_*$ and construct counterexamples in $n>n_*$ by perturbing the noncompactness examples. To the best of the author's knowledge, these are the first locally conformally flat noncompactness constructions for the positive-scalar-curvature, minimal-boundary case of the Yamabe--Escobar equation, outside the conformal class of the round hemisphere.
Comments103 pages
