AI 中文总结
该研究针对$3\leq N\leq5$的双曲区域上的临界Dirichlet问题，利用拓扑与解析方法证明：当区域满足特定同调非平凡条件时，非平凡拓扑可克服紧性损失与几何位势，迫使正解存在。
AI 中文摘要
设$\Omega\Subset\mathbb H^N$（$N\in\{3,4,5\}$）为有界连通$C^2$区域，我们证明：若对某个$1\le d\le N-1$有$H_d(\Omega;\mathbb F_2)\neq0$，则纯临界Dirichlet问题$-\Delta_{\mathbb H}u=u^{\frac{N+2}{N-2}}$在$\Omega$内成立，且在$\partial\Omega$上$u=0$，此时该问题存在正解。这给出了$3\le N\le5$下$C^2$边界正则性的双曲Bahri-Coron定理。在共形约化下，双曲几何产生正位势并引出依赖维度的气泡机制：三维情形下，所需能量下降源于固定重数下对角修正与对相互作用的平衡；四维和五维情形下，通过归一化缺陷估计和全对源转移界在匹配尺度上获得能量下降。统一的Thom重心构造将这些解析估计转化为拓扑矛盾，因此尽管存在临界紧性损失和额外几何位势，非平凡区域拓扑仍迫使解存在。
英文摘要
Let \(Ω\Subset\mathbb H^N\), \(N\in\{3,4,5\}\), be a bounded connected \(C^2\) domain. We prove that the pure critical Dirichlet problem \[ -Δ_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }Ω, \qquad u=0 \quad\text{on }\partialΩ \] admits a positive solution whenever \(H_d(Ω;\mathbb F_2)\neq0\) for some \(1\le d\le N-1\). This gives a hyperbolic Bahri-Coron theorem for \(3\le N\le5\) under \(C^2\) boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.
Comments102 pages, comments are welcome
