发表机构
Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用支撑函数表示研究双曲和de Sitter平面上的超定方程，证明椭圆情形下常数边界数据迫使测地圆盘，而双曲情形下局部刚性失效，揭示了从椭圆刚性到双曲灵活性的转变。
AI 中文摘要
我们通过Lorentz-Minkowski三维空间中零平均曲率曲面的支撑函数表示，研究双曲平面和de Sitter平面上的超定方程。在双曲平面上，方程$\Delta u-2u=0$的解产生带分支的类空极大曲面。我们证明，在有界单连通区域上，非平凡的常数Dirichlet和Neumann数据迫使该区域为测地圆盘，前提是相关的支撑二次曲面是非类光的；此外，解是到中心距离的双曲余弦的倍数。在de Sitter平面上，相应的方程为Klein-Gordon方程$\Box u+2u=0$，其解在支撑张量非退化处生成类时极小曲面。我们确定了由常数Cauchy数据决定的支撑二次曲面和常角条件，证明了沿每条解析非特征曲线的局部刚性不成立，并从全局Cauchy圆上的常数数据恢复旋转对称性。因此，相同的支撑函数形式揭示了当符号变化时从椭圆刚性到双曲灵活性的显著转变。
英文摘要
We study overdetermined equations on the hyperbolic and de Sitter planes by means of the support-function representation of zero mean curvature surfaces in Lorentz--Minkowski three-space. On the hyperbolic plane, solutions of $Δu-2u=0$ give rise to branched spacelike maximal surfaces. We prove that, on a bounded simply connected domain, nontrivial constant Dirichlet and Neumann data force the domain to be a geodesic disk, provided that the associated support quadric is nonlightlike; moreover, the solution is a multiple of the hyperbolic cosine of the distance from the center. On the de Sitter plane, the corresponding equation is the Klein--Gordon equation $\Box u+2u=0$, and its solutions generate timelike minimal surfaces wherever the support tensor is nondegenerate. We identify the support quadric and the constant-angle condition determined by constant Cauchy data, prove that local rigidity fails along every analytic noncharacteristic curve, and recover rotational symmetry from constant data on a global Cauchy circle. Thus the same support-function formalism reveals a sharp transition from elliptic rigidity to hyperbolic flexibility when the signature changes.
Comments13 pages. Comments are welcome!
