Approaching the prescribed Gaussian curvature by discrete conformality
Ziran Liu, Tianqi Wu
TL;DR
This work tackles the classical problem of prescribing Gaussian curvature on 2D manifolds by developing a discrete conformality framework based on vertex scaling on geodesic triangulations. It proves that, for genus $>1$ surfaces and negative target curvature, there exists a unique discrete conformal factor $u$ solving $K(u)=0$, found by minimizing a globally convex functional and refined through an ODE-based deformation that preserves a positive metric and yields an $O(|l|)$-accurate approximation to the smooth solution as the triangulation becomes finer. The methodology hinges on a precise discrete calculus on graphs, an explicit formula for the Jacobian $\frac{\partial K}{\partial u}(u)=D(u)-\Delta_{\eta(u)}$, and key geometric/elliptic estimates to guarantee convergence under $\epsilon$-acute triangulations with small edge lengths. The approach provides an efficient computational route to discrete prescribed-curvature problems on high-genus surfaces, leveraging convex optimization and careful discrete-to-smooth convergence analysis.
Abstract
We propose a discrete approach for approximating solutions to the prescribed Gaussian curvature problem in two-dimensional manifolds, based on the notion of discrete conformality. Our approach provides an efficient numerical method to compute the solution by minimizing a convex functional.