边界 Yamabe 问题中紧致性的临界维数，I
On critical dimensions for compactness in the boundary Yamabe problem, I
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中文总结 AI 辅助
本文构造了边界 Yamabe 方程具有无界正解序列的光滑非局部共形平坦度量，确定了不同曲率条件下紧致性失效的临界维数范围，并证明了约化能量二次项具有负的非退化局部极值。
中文摘要 AI 辅助
我们在闭球上构造了光滑、非局部共形平坦的度量，使得边界 Yamabe 方程允许 $L^\infty$-无界的正解序列。沿每条序列，背景度量以及给定的标量曲率和边界平均曲率是固定的。对于零标量曲率和正边界平均曲率，在脐边界且维数 $N\ge22$ 以及非脐边界且 $N\ge15$ 的所有维数中，都存在这样的例子。对于正标量曲率和最小边界，相应的范围是 $N\ge21$ 和 $N\ge15$。对于每个 $N\ge9$，当标量曲率为 $N(N-1)$ 且给定的边界平均曲率低于依赖于 $N$ 的负阈值时，也存在具有脐或非脐边界的例子。该构造使用半空间上的多项式度量扰动和修正气泡，然后通过共形紧化到球。我们评估或估计求解线性化 Neumann 或 Robin 边界问题的修正及其对约化能量二次项的贡献。我们证明该二次项在所述所有维数中，关于切向平移和尺度具有负的、非退化的局部极值。
英文摘要
We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits $L^\infty$-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension $N\ge22$ and with nonumbilic boundary in every $N\ge15$. For positive scalar curvature and minimal boundary, the corresponding ranges are $N\ge21$ and $N\ge15$. For every $N\ge9$, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is $N(N-1)$ and the prescribed boundary mean curvature is below a negative threshold depending on $N$. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.
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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