边界 Yamabe 问题中紧致性的临界维数，II
On Critical Dimensions for Compactness in the Boundary Yamabe Problem, II
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中文总结 AI 辅助
本文确定了边界 Yamabe 问题中紧致性的临界维数，给出不同边界条件下的尖锐范围，并通过标量修正与几何公式证明过渡维数。
中文摘要 AI 辅助
我们确定了正共形类型的光滑紧致流形上标量平坦和极小边界 Yamabe 问题的尖锐紧致性范围，排除了共形圆半球。对于零标量曲率和正常数边界平均曲率，紧致性在一般边界情形下成立至维数 14，在脐边界情形下成立至维数 21。对于正标量曲率和零边界平均曲率，相应的上界维数分别为 14 和 20。结合第一部分中的非紧致性例子，这些结果确定了两类边界中的过渡维数。对于正标量曲率，我们还证明了对于每个固定的实数边界平均曲率，紧致性成立至维数八，并且当该曲率接近零或足够大且为正时，获得了更高维数的范围。证明结合了全共形 Fermi 度量展开的标量修正估计与一个几何公式，该公式将修正能量的对数系数表示为平方的负和。
英文摘要
We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for general boundary and dimension $21$ for umbilic boundary. For positive scalar curvature and zero boundary mean curvature, the corresponding upper dimensions are $14$ and $20$. Together with the noncompactness examples in Part I, these results identify the transition dimensions in both boundary classes. For positive scalar curvature, we also prove compactness through dimension eight for every fixed real boundary mean curvature, and obtain higher-dimensional ranges when this curvature is near zero or sufficiently large and positive. The proof combines scalar-correction estimates for the full conformal Fermi metric expansion with a geometric formula expressing the logarithmic coefficient of the corrected energy as a negative sum of squares.
发表机构
- Chinese University of Hong Kong（香港中文大学）
- Hanyang University（汉阳大学）
- University of Bath（巴斯大学）
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