Variational principles and combinatorial \(p\)-th Yamabe flows on surfaces
The authors start giving details on PL-metrics (piecewise linear metrics) in Euclidean triangulated manifolds. For these types of metrics, the most natural curvature is the combinatorial Gauss curvature. Considering the combinatorial Gauss-Bonnet formula associated with this curvature one obtains the average of the total combinatorial Gauss curvature designated by \(K_{av}\). The constant combinatorial PL-metric has curvature \(K_{av}\) at all vertices. The Yamabe combinatorial problem deals with the existence of this metric. The authors start by explaining previous studies on this problem done by other mathematicians. \textit{F. Luo} [Commun. Contemp. Math. 6, No. 5, 765--780 (2004; Zbl 1075.53063)] studied the discrete Yamabe problem through the combinatorial Yamabe flow. Luo showed that the combinatorial Yamabe flow is the negative gradient of a potential functional \(F\). He showed that \(F\) is locally convex and obtained the local rigidity for the curvature map and the local convergence of the combinatorial Yamabe flow. He also conjectured that the combinatorial Yamabe flow converges to a constant curvature PL-metric after a finite number of surgeries on the triangulation. \textit{H. Ge} and \textit{W. Jiang} [Calc. Var. Partial Differ. Equ. 55, No. 6, Paper No. 136, 14 p. (2016; Zbl 1359.53054)] introduced the extended combinatorial Yamabe algorithm to handle possible singularities along the combinatorial Yamabe flow and \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] started doing surgery by ``flipping the algorithm''. In this paper the authors generalize results for both extended combinatorial Yamabe flow and combinatorial Yamabe flow with surgery. The generalization is done first for \(p>1\) introducing the extended combinatorial \(p\)-th Yamabe flow, which is the extended Yamabe flow when \(p=2\) introduced by Ge and Jiang and then generalize the main results they obtained. They show that the solution to the extended combinatorial \(p\)-th Yamabe flow exists for all time. The details are all explained in Section 2. In Section 3 the authors recall the discrete conformal theory and discrete unifomization theorem established by \textit{X. D. Gu} et al. [J. Differ. Geom. 109, No. 2, 223--256 (2018; Zbl 1396.30008); J. Differ. Geom. 109, No. 3, 431--466 (2018; Zbl 1401.30048)] and generalize their results for the combinatorial \(p\)-th Yamabe flow with surgery. It is shown that for the generalized \(p\)-th flows \(p>1\) and \(p\neq 2\) there exists only curvature convergence but no exponential convergence as in the case of \(p=2\).
scientific article; zbMATH DE number 7450416
scientific article