2025/02/03 by Xiaolong Han, Han, Xiaolong Hans
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Geodesic #Geology #Geometry #Mathematics
paper · pdf · doi:10.48550/arxiv.2502.01134
openalex publication_date 2025/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a closed hyperbolic 3-manifold. A homotopy class [S] of surfaces in M is filling if any representative cuts M into components contractible in M. We prove that there exist ε0, g0>0 such that every homotopy class of (1+ε)-quasi-Fuchsian surfaces with 0<ε≤ ε0 or totally geodesic surfaces of genus ≥ g0 in M is filling. As a corollary, except for at most finitely many totally geodesic surfaces, embedded incompressible quasi-Fuchsian surfaces in M have constants bounded below by 1+ε0. This also gives a gap theorem for embedded minimal surfaces. Each of these surfaces separates any pair of distinct points at the sphere of infinity. Crucial tools include the rigidity results of Mozes-Shah, Ratner, and Shah. This work is inspired by a question of Wu and Xue whether random geodesics on random hyperbolic surfaces are filling.