论文标题
在最低能级的Yamabe方程的签名爆炸
Sign-changing blow-up for the Yamabe equation at the lowest energy level
论文作者
论文摘要
我们研究了Yamabe类型的Riemannian歧管$（m，g）$上Yamabe方程的标志溶液序列的爆炸行为。对于每个维度$ n \ ge11 $，我们描述发生爆炸的最小能量阈值的值。在$ 11 \ le n \ le 24 $中，已知正面解决方案的集合是紧凑的，我们表明一组改变签名解决方案不是紧凑的，并且爆破已经发生在最低的能量水平上。我们通过在空间形式上构建平滑的，非局部的共同扁平度度量来证明这一结果，$ \ mathbb {s}^n/γ$，$γ\ neq \ neq \ {1 \} $，其Yamabe方程接纳了一个变化的爆炸解决方案。作为此结果的对应物，我们还证明，在最低能级，小维度或强烈的几何假设下，签名换解决方案的急剧紧凑性结果。
We investigate the blow-up behavior of sequences of sign-changing solutions for the Yamabe equation on a Riemannian manifold $(M,g)$ of positive Yamabe type. For each dimension $n\ge11$, we describe the value of the minimal energy threshold at which blow-up occurs. In dimensions $11 \le n \le 24$, where the set of positive solutions is known to be compact, we show that the set of sign-changing solutions is not compact and that blow-up already occurs at the lowest possible energy level. We prove this result by constructing a smooth, non-locally conformally flat metric on space forms $\mathbb{S}^n/Γ$, $Γ\neq \{1\}$, whose Yamabe equation admits a family of sign-changing blowing-up solutions. As a counterpart of this result, we also prove a sharp compactness result for sign-changing solutions at the lowest energy level, in small dimensions or under strong geometric assumptions.
