Guided by the Contact Polygon Method, we establish a topological design principle for constructing periodically dome-shaped graphene (PDG). Integrating different membered non-hexagonal carbon rings into honeycomb frameworks of the PDGs induces spontaneous surface positive or negative Gaussian curvature, with the former yielding tunable Poisson’s ratios for small membered rings and the latter dictating strong full negative Poisson’s ratio behavior for large membered rings. Furthermore, the PDG with five-membered rings exhibits both a nearly full zero Poisson’s ratio and superhigh electron mobility of 91 × 10^4 cm^2V{-1}s^{-1} which is three times that of graphene. Building upon these PDGs, structural modifications are introduced to extend their functionalities. Modified PDG with mirror symmetry plane and sp^2-sp3 carbon bonding is found to suppress lattice thermal conductivity of 2D carbon networks to a new low level (40 Wm^{-1}K{-1}) via low-lying optical-acoustic phonon intersections and large acoustic phonon softening, while BN co-doped PDG with a composition of C2BN demonstrates a superconducting transition temperature of 22 K fueled by a Fermi-level Van Hove singularity. This study establishes a new bridge between structural topology and advanced functionalities of multifunctional 2D carbon networks.