发表机构
University of Rochester; University of Florida（罗切斯特大学; 佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对闭黎曼曲面上带变号系数的奇异Liouville系统，通过识别参数空间中的内在射线将系统约化为标量变号平均场方程，结合变分理论与紧性定理，证明了该射线上所有非临界参数处同步解的存在性。
AI 中文摘要
我们考虑闭黎曼曲面上的奇异Liouville系统，其系数为公共的变号函数。从正系数出发的直接延拓因同伦过程中不可避免出现退化零点而受阻。我们转而识别参数空间中的一条内在射线，在该射线上系统允许同步解，并精确约化为一个标量变号平均场方程。将此约化与De Marchis--López-Soriano--Ruiz的变分存在性理论以及Wu的节点奇点紧性定理相结合，我们获得了该射线上每个非临界参数的存在性。在节点曲线上允许正奇异源。特别地，当曲面具有非正欧拉示性数且系数的每个负分量都是圆盘时，标量极小极大构造所需的拓扑假设自动满足。
英文摘要
We consider singular Liouville systems on closed Riemann surfaces with a common sign-changing coefficient. Direct continuation from a positive coefficient is obstructed by the unavoidable appearance of degenerate zeroes along such a homotopy. We instead identify an intrinsic ray in the parameter space on which the system admits synchronized solutions and reduces exactly to a scalar sign-changing mean-field equation. Combining this reduction with the variational existence theory of De Marchis--López-Soriano--Ruiz and the nodal singularity compactness theorem of Wu, we obtain existence for every noncritical parameter on this ray. Positive singular sources are allowed on the nodal curve. In particular, when the surface has nonpositive Euler characteristic and every negative component of the coefficient is a disk, the topological hypothesis required by the scalar min--max construction is automatic.
Comments9 pages
