AI 中文总结
该研究通过环境对称性，结合rim surgery、凸几何与双曲几何，构造4维流形中的射影刚性、拓扑灵活曲面及全测地曲面，揭示奇异纽结与精细对称性现象。
AI 中文摘要
我们通过4维流形的环境对称性研究其中曲面的奇异纽结。首先给出生成射影刚性曲面的通用方法，这类曲面的每个光滑可延拓自微分同胚在第一同调群上的作用为恒等或负恒等。对每个整数g>0，该构造的细化版本会在4维流形X_g中生成亏格为g的曲面序列F_0,…,F_2g。这些曲面是拓扑同痕且拓扑灵活的：F_i的每个保定向自微分同胚都可由X_g的保F_i自同胚实现。然而，连续纽结会排除越来越多的射影同调对称性，揭示出更精细的纽结现象。前两种构造结合了迭代 rim surgery 与相对 Seiberg-Witten 不变量的牛顿多面体凸几何。我们还利用双曲几何构造了一个亏格为正的全测地曲面，其光滑可延拓和拓扑可延拓的映射类群均为平凡群。
英文摘要
We study exotic knottings of surfaces in 4-manifolds through their ambient symmetries. We first give a general recipe for producing projectively rigid surfaces, for which every smoothly extendable self-diffeomorphism acts on first homology by plus or minus the identity. For every integer g >0, a refinement of this construction yields a finite sequence of genus-g surfaces F_0, ..., F_2g contained in a 4-manifold X_g. These surfaces are topologically isotopic and topologically flexible: every orientation-preserving self-diffeomorphism of F_i can be realized by a self-homeomorphism of X_g preserving F_i. Successive knotting, however, rules out increasingly many projective homological symmetries, revealing a finer knottedness phenomenon. The first two constructions combine iterated rim surgery with the convex geometry of Newton polytopes of relative Seiberg-Witten invariants. We also use hyperbolic geometry to construct a totally geodesic surface of positive genus whose smooth and topological extendable mapping class groups are both trivial.
Comments23 pages
