Yamabe稳定性优化元的存在性
Existence of Yamabe stability optimizers
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中文总结 AI 辅助
该研究证明了满足特定阈值条件的高维正Yamabe不变量闭Riemann流形上Yamabe不等式稳定性优化元的存在性，明确了不同情形下的紧性阈值与严格不等式成立的条件。
中文摘要 AI 辅助
我们证明了，在维数至少为3且具有正Yamabe不变量、满足两个阈值条件的闭Riemann流形上，Yamabe不等式的稳定性优化元存在。值得注意的是，我们发现的紧性阈值不同于第二作者之前研究的圆球面特殊情形，更准确地说，该阈值由以1个而非2个气泡爆破的序列给出，反映了非球面情形下Yamabe极小化元的紧性。利用Aubin–Schoen测试函数的经典渐近分析，我们证明，在维数至少为6且流形非局部共形平坦时，稳定性常数严格低于单气泡阈值；在互补情形，即维数3至5或流形局部共形平坦时，我们得到一个新的正质量型条件，该条件是严格不等式成立的充分条件。
英文摘要
We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin--Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.
