AI 中文总结
本文研究带边界权重的重心空间，将其作为非紧边界欧拉-拉格朗日泛函集中模式的拓扑模型，通过代数拓扑处理推导相关性质，并应用于带边界曲面与半球的平均场方程拓扑研究。
AI 中文摘要
我们研究带边界权重的重心空间，其中内部支撑点的代价（或权重）为2，边界支撑点的代价为1。这些空间是由非紧边界欧拉-拉格朗日泛函产生的集中模式的有限维拓扑模型：内部气泡携带的量化质量是边界气泡的两倍，因此非常负的亚水平集由带边界权重的重心而非普通重心建模。本文对这些空间进行了系统的代数拓扑处理。我们构造了混合层、闭层偏序集以及三角带边界权重的余极限滤子；计算了欧拉示性数；并根据∂M和M/∂M的普通重心空间证明了同调分解。随后，我们将公式专门应用于带边界的紧连通可定向曲面和半球，得到了显式有理贝蒂多项式，对于曲面，还得到了平均场应用所需的模2多项式。最后，我们解释了这些多项式如何度量带边界紧曲面上共振诺伊曼平均场方程的无穷远处拓扑。
英文摘要
We study boundary-weighted barycenter spaces, in which an interior support point has cost (or weight) two and a boundary support point has cost one. These spaces are finite-dimensional topological models for the concentration patterns produced by noncompact boundary Euler--Lagrange functionals: an interior bubble carries twice the quantized mass of a boundary bubble, and very negative sublevels are therefore modeled by boundary-weighted rather than ordinary barycenters. The paper gives a systematic algebraic-topological treatment of these spaces. We construct the mixed strata, the closed-stratum poset, and the triangular boundary-weighted colimit filtration; compute the Euler characteristic; and prove a homology decomposition in terms of ordinary barycenter spaces of \(\partial M\) and \(M/\partial M\). We then specialize the formulas to compact connected orientable surfaces with boundary and to hemispheres, obtaining explicit rational Betti polynomials and, for surfaces, the mod-two polynomials required by the mean-field application. Finally, we explain how these polynomials measure the topology at infinity in the resonant Neumann mean-field equation on a compact surface with boundary.
