发表机构
East China Normal University; Zhejiang Normal University（华东师范大学; 浙江师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明双调和方程正破裂解在曲率非负或无穷远增长条件下必为径向对称的尖锐破裂解，并构造反例说明条件必要性。
AI 中文摘要
我们建立了在$\mathbb R^3\setminus\{0\}$中共形不变方程$\Delta^2 u=u^{-7}$的正破裂解的两个刚性定理，这些解连续延拓到原点且满足$u(0)=0$。首先，我们证明若相关的共形度量$g=u^{-4}|dx|^2$具有非负数量曲率，则每个这样的解都是径向对称的，并具有尖锐破裂轮廓$u(x)\sim(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$当$x\to 0$时；特别地，该度量在原点处是完备的。主要新颖之处在于一个既不需要曲率也不需要对称性的全局刚性定理：单一全局条件$u(x)=o(|x|)$在无穷远处迫使$u(x)\equiv(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$。这一结果对McKenna和Reichel在无先验对称性假设的类中提出的唯一性问题给出了尖锐回答。最后，我们构造了互补的例子，证明曲率和增长假设的关键作用。
英文摘要
We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation $Δ^2 u=u^{-7}$ in $\mathbb R^3\setminus\{0\}$, which extend continuously to the origin with $u(0)=0$. First, we prove that if the associated conformal metric $g=u^{-4}|dx|^2$ has nonnegative scalar curvature, then every such solution is radially symmetric and has the sharp rupture profile $u(x)\sim(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$ as $x\to 0$; in particular, the metric is complete at the origin. The principal novelty is a global rigidity theorem requiring neither curvature nor symmetry: the single global condition $u(x)=o(|x|)$ at infinity forces $u(x)\equiv(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$. This result provides a sharp answer to the uniqueness question posed by McKenna and Reichel in a class with no a priori symmetry assumption. Finally, we construct complementary examples demonstrating the essential roles of the curvature and growth hypotheses.
Comments23 pages
