Global existence and convergence of authalic flows

Establish global existence of the continuous authalic flow and global convergence of the discrete authalic-flow algorithms, and determine how multiple critical points of the discrete stretch energy affect the degree of area preservation achieved by the resulting maps.

Background

The paper introduces the authalic flow as the continuous L2L^2-gradient flow of the stretch energy for diffeomorphisms between equal-area Riemannian 2-manifolds, and develops a discrete authalic-flow algorithm for simplicial maps. The analysis establishes monotonic decrease of the continuous stretch energy conditional on the existence of a smooth, orientation-preserving diffeomorphic solution, while the discrete theory proves consistency and an area-distortion estimate for global minimizers. It does not establish global-in-time existence of the continuous flow or convergence of the numerical algorithms over arbitrary initial data and parameterizations.

The authors also note that the discrete stretch energy may have multiple critical points. Consequently, different converged maps may exhibit different levels of area preservation, motivating the unresolved problem of characterizing the critical-point structure and its effect on the output of the discrete method.

References

That said, the proposed framework of authalic flow has several limitations. Global existence of the continuous authalic flow and global convergence of discrete algorithms have not yet been established. Moreover, the discrete stretch energy may admit multiple critical points, which can lead to varying degrees of area preservation in the resulting maps. We leave these questions for future work.

— Area-Preserving Parameterization: Variational Principle, Gradient Flow, and Discrete Approximation  (2608.17873 - Liu et al., 18 Aug 2026) in Section 7, Conclusion and discussion